The outgoing president said a new election would help preserve "peace and consensus" following a week of street protests by supporters of opposition candidate Viktor Yushchenko.
Earlier yesterday, Kuchma criticized protesters in Kyiv for blocking access to the central bank and other public buildings, saying such actions were threatening the country's economy.
"A few more days and our financial system will fall apart like a house of cards, and neither the president nor the government can be held responsible for that," Kuchma said. "We cannot even work normally."
Yushchenko, who rejects official results that gave Prime Minister Viktor Yanukovych victory in the runoff, did not react to Kuchma's proposal to hold a new ballot.
Instead, Yushchenko said he had great hopes on the Verkhovna Rada, the Ukrainian parliament, which on 27 November passed a nonbinding resolution pronouncing the election results flawed and against the will of the Ukrainian people. The legislature today started debating a no-confidence motion against the Yanukovych government.
Yushchenko yesterday made it clear that such a motion would be a symbolic victory for the opposition, even if had no legal effect.
"We expect the parliament to decide tomorrow [30 November] on the issue of the government's actions," Yushchenko said. "I think that tomorrow we will obtain the dismissal of Yanukovych and his government. We will also obtain the removal of those heads of eastern regional administrations who have suggested that Ukraine should be dismembered. I think the dismissal of Prosecutor-General [Hennadiy Vasiliev] will be also announced tomorrow."
The parliamentary session, however, was adjourned after a few hours amid disagreement over the content of the planned resolution. The debate is expected to resume tomorrow.
Kuchma's proposals were overshadowed by the resumption today of Supreme Court hearings on complaints of election fraud filed by the opposition.
Under Ukrainian laws, the Supreme Court cannot rule on the overall results of the 21 November polls. It can, however, invalidate returns from individual polling stations.
The opposition's appeal, which began yesterday, focuses on results filed by a number of regions -- including Kyiv, Odesa, and Nikolayevsk.
Oleksandr Litvinenko is a political and legal expert at the Kyiv-based Ukrainian Center for Economic and Political Studies, also known as the Oleksandr Razumkov Center. He told RFE/RL that under Ukrainian law, it is up to the legislature to decide whether voters should go to the polls again.
To what extent the Supreme Court's ruling could influence a decision by parliament is unclear.
As for Kuchma's proposal to hold a new election, Litvinenko said that in addition to having no legal effect, it is unlikely to suit his opponents.
"This is not the only possibility [that could help resolve the current crisis] and, besides, the opposition is rather leaning toward a repeat of the second round only," Litvinenko said. "[They say] it would be simpler, quicker, and cheaper."
A repeat of the 21 November runoff would pit Yushchenko against Yanukovych. But in the event of an entirely new vote, things are less clear-cut and it is unclear whether either of the two finalists would be able to run again.
Even before Kuchma said he was in favor of a new election, some experts had evoked such a scenario as part of a possible move to sideline Yushchenko. Litvinenko said such a possibility might still appeal to the Ukrainian leadership.
"It is difficult to answer this question by 'yes' or 'no' with hundred-percent certainty, but, surely enough, several options are being considered. When the Verkhovna Rada adopted its resolution [on 27 November], the Communists and their leader, Petro Symonenko, demanded that neither Yanukovych nor Yushchenko be allowed to run for president again. It is therefore obvious that some political forces are considering such a scenario that would sideline both candidates in the event of a new election."
Yanukovych today offered his rival the post of prime minister in a coalition government. He also said that in case the outcome of the polls were to be invalidated, he would agree to a new vote if neither he nor Yushchenko ran.
He said either option would help avoid a split in the country.
Yushchenko and his allies have not reacted to Yanukovych's statement.
Ukrainian political expert and civil rights activist Roman Romanov said the opposition is unlikely to accept any compromise.
"I believe the authorities had been keeping several scenarios in reserve and are now bringing this one into life," Romanov said. "Under such a scenario, the election would be invalidated and the authorities would nominate a candidate that would perhaps be less tainted than Yanukovych and would try to have the opposition seek another leader -- which seems unlikely at such short notice. This is why I do not believe this scheme will bring the expected result."
Meanwhile, Serhiy Tyhypko, the former head of Yanukovych's campaign staff, suggested today that he might bid for the presidency in the event of a new election.
A former deputy prime minister, Tyhypko chairs the pro-government Labor Ukraine Party. He resigned yesterday from his post of central-bank chief, saying he intended to devote his time solely to politics.
Some analysts had tipped Tyhypko as a potential government candidate before the election campaign started. But Kuchma instead picked Yanukovych -- a little-known figure despite his status of prime minister -- as his heir apparent.
Political expert Romanov said he believes Tyhypko would suit the outgoing Ukrainian leader in the event of a revote. But he said the move could prove risky.
"Such an option would certainly be rejected by the Ukrainian people, because it is precisely against this political system -- rather than against Yanukovych in particular -- that they are protesting," Romanov said.
[Click here for more RFE/RL coverage of Ukraine's disputed presidential election.]