The demonstrators are calling upon policemen on guard near the building to join them.
An RFE/RL correspondent at the scene reported seeing some of the protesters shaking the wrought-iron fence that surrounds the official compound.
The number of protesters swelled after Bakiev earlier today submitted to parliament his own proposals to amend the constitution.
Bakiev's aides say the proposals aim at giving the legislature more powers, but the opposition accuses the head of state of "cheating."
Meanwhile, Bakiev today removed acting Interior Minister Osmanali Guronov from office.
The head of the presidential administration, Myktybek Abdyldaev, appeared before protesters picketing the government headquarters in Bishkek and read out a presidential decree putting Omurbek Subanaliev temporarily in charge of the ministry. The reasons for the reshuffle were not immediately clear. Demands For Parliamentary System
State Secretary Adakhan Madumarov told reporters that Bakiev is proposing to give parliament more prerogatives, including those of appointing the prime minister. He said that Bakiev's initiative aims at creating a mixed presidential-parliamentary form of government.
But opposition leaders say they want a parliamentary form of government and warn that Bakiev's proposals will only make the crisis worse.
Protests have been going on since November 2 and organizers have said that they will continue until their demands are met.
THE COMPLETE STORY:
Click on the icon to view a dedicated webpage bringing together all of RFE/RL's coverage of Kazakhstan, Kyrgyzstan, Tajikistan, Turkmenistan, and Uzbekistan.