In an address to the country on January 20, Yanukovych called the violence a threat to all of Ukraine.
He pledged to ensure the rights of demonstrators to legal protests and also to ensure security.
"I appeal to Kyiv citizens and other participants of mass rallies -- I ask you not to follow those who call for violence, who are willing to provoke the split between the state and society, who want to throw the people of Ukraine into the furnace of mass riots," Yanukovych said.
International leaders and rights groups have urged both sides to deescalate the situation and avoid bloodshed.
The unrest follows the passage and enactment in the past week, following nearly two months of street protests, of new legislation in Ukraine that dramatically curbs the right to demonstrate and imposes harsh penalties for broadly defined "extremist activity."
Yanukovych said in his address: "I have treated your participation in mass rallies with understanding; I have expressed readiness to listen to your position and jointly find ways of solving the existing contradictions. However, now, when peaceful actions have escalated into mass riots accompanied by demolition, arson, and violence, I am confident that such phenomena threaten not only the safety of the residents of Kyiv but the whole Ukraine."
WATCH: RFE/RL Ukrainian Service live video as protesters and police engage in a tense standoff in Kyiv:
Ukraine's prosecutor-general, Viktor Pshonka, meanwhile warned protesters against what he called "mass rioting," calling it a "crime against the state."
Earlier on January 20, Yanukovych agreed to talks with the opposition after violent clashes between antigovernment protesters and riot police in Kyiv.
Yanukovych said he has set up a special committee -- headed by national security chief Andriy Kluyev -- to meet with opposition representatives to work out a solution to the crisis.
On January 19, antigovernment protesters battled security forces in what has been described as the worst violence since protests against Yanukovych's government began after Kyiv turned its back on talks over an EU Association Agreement two months ago.
Sporadic clashes between protesters and riot police continued overnight, and on January 20 dozens of protesters remained camped out on a central street in Kyiv.
More than 200 people, about half of them protesters, were injured in the clashes. The city health department said 103 protesters were treated for injuries, 42 of whom were hospitalized. The Interior Ministry said about 100 police officers were injured, of whom 61 were hospitalized.
The Interior Ministry said more than 20 people have been detained.
RFE/RL correspondents Dmytro Barkar and Ihor Iskhakov were also detained on January 20 while they were providing live video coverage of the clashes.
Both journalists were released hours later and said they had been beaten by police. Their faces showed bruises and scratches. Iskhakov said his nose may have been broken and he also might have a concussion.
WATCH: RFE/RL journalist Dmytro Barkar and cameraman Ihor Iskhakov describe their treatment at the hands of police when they were being detained on January 20:
The journalists said the videocamera they were using is missing.
Police spokeswoman Inna Hamaliy told RFE/RL that Iskhakov and Barkar were detained as "witnesses."
Earlier on January 19, tens of thousands of people rallied on Kyiv's Independence Square against tough new antiprotest legislation.
SLIDESHOW: Kyiv protests turn violent
U.S.-based Human Rights Watch (HRW) said it was concerned the government could use the outbreak of violence by a minority of demonstrators to crack down on the majority of peaceful protesters.
"Our principle concern is that neither the police nor the government use the outbreak of violence by a tiny minority of the demonstrators as an excuse to crack down on the vast majority of the demonstrators who continue to protest peacefully," HRW Executive Director Kenneth Roth told Reuters.
Tensions soared after the Verkovna Rada, Ukraine's parliament, approved sweeping legislation on January 16 that significantly increased fines and imposed jail terms for unauthorized street protests.
The legislation was approved by hand vote by the ruling parties without the presence of the opposition.
The measures, signed into law by Yanukovych, also amended the criminal code to introduce provisions recriminalizing defamation. The measures also restrict the activity of nongovernmental organizations funded by the West, as many are in Ukraine.
The antigovernment protests were sparked by Yanukovych's surprise decision in November to pull out of a landmark treaty with the European Union.
Pro-EU protesters who have been occupying Independence Square since then are demanding Yanukovych's resignation.
The opposition confirmed Yanukovych's offer of talks, which came after opposition leader Vitali Klitschko met with Yanukovych at his presidential residence outside Kyiv.
Opposition leader Arseniy Yatsenyuk told protesters what they would be demanding, saying that Yanukovych's government had declared "that they are willing to start a negotiation process."
Key Points Of Ukraine's Antiprotest LawsKey Points Of Ukraine's Antiprotest Laws
The sweeping legislation:
* Introduces criminal responsibility for libel and further criminalizes "extremist activity." For instance, the production or sharing, including through media, of "extremist papers" that might involve calling for "mass riots" may lead to a fine or up to three years' imprisonment.
* Makes it an offense to wear a mask or helmet at a public rally. Doing so could lead to a fine of up to $520 or up to 15 days' detention.
* Outlaws protest motorcades of more than five cars without police permission. Punishment here ranges from a fine of up to $100 to confiscation of the car and suspension of a driving license for up to two years.
* Bans the unauthorized setting up of stages or tents during public rallies, punishable by a fine of up to $620 or up to 15 days' detention.
* Introduces harsh criminal penalties for certain violations. For example, blocking access to living quarters is an offense punishable by up to six years in prison.
* Requires all mass media providing information services to the public to register as an "information agency." Internet users in Ukraine may be banned from the Internet based on an expert decision that they disseminate criminal information.
* Requires nongovernmental organizations that receive funding from foreign sources and participate in "political activities" to register as "foreign agents." Such groups must include the words "foreign agent" in their names and all their materials, and are required to pay income tax. "Foreign agents" will have three months to register, and failure to comply may lead to the organization being disbanded.
* Enables parliament to strip deputies of immunity without pre-approval by a special parliamentary committee.
"The secretary of the council of national security and defense Andriy Kluyev is responsible for these negotiations," he added. "I can tell you my personal vision -- first, the mandate for negotiations is only the mandate of the Maidan [as the protests have become known], second -- exclusively public negotiations, and third -- our negotiation is the fulfillment of our demands."
Ukrainian media said on January 20 that a radical right-wing youth group was behind the clashes with police during the previous day's protests, when hundreds of demonstrators tried to storm a police cordon near the parliament building, setting police buses and other vehicles on fire. Police responded with tear gas and water cannon.
The United States has urged an end to the violence. On January 19, U.S. National Security Council spokeswoman Caitlin Hayden also called on Ukraine to repeal recent laws limiting protests, to remove riot police from downtown Kyiv, and to start talking to the opposition.
The European Union also condemned the violence, and on January 20, EU foreign policy chief Catherine Ashton voiced hope that the proposed talks would be a step toward resolving the political crisis.
"I have been following very closely the events in Ukraine during the last few days," she said. "I see that [Ukrainian] President [Viktor] Yanukovych will be talking today with opposition leaders. I hope that that is a positive move forward."
Swedish Foreign Minister Carl Bildt welcomed the proposed talks but said a solution is only possible if there is no regime repression.
EU foreign ministers are to discuss the situation in Ukraine at a meeting in Brussels on January 20.