Opposition had demanded that the president either resign, or agree to a new Constitution slashing his powers.
Protests in central Bishkek turned violent outside the government building on Tuesday: six people were hospitalized with injuries and two were reported to be suffering from tear gas poisoning after riot police dispersed opposition demonstrators following five days of protests.
However, Azimbek Beknazarov, a member of the opposition movement For Reform, said later in the day that the sides had reached an agreement on constitutional reform, and opposition leaders intended to draw the rally to a close.
"We are talking to our colleagues and persuading them to stop the rally. We must thank the people who have supported us, and convince them to leave [the central square]," Beknazarov said.
Bakiyev told his Russian counterpart Vladimir Putin in a phone conversation on Tuesday that his country's authorities had the situation in the Central Asian republic under control.
Many protesters remain on the central square, pledging to stay until the constitutional agreement is implemented.
Under the new draft Constitution, the president will lose his right to dissolve parliament, and the legislature will have the right to appoint the prime minister and cabinet.
The opposition has accused President Bakiyev and the government of failing to deliver on promises to crack down on corruption and raise living standards in the impoverished ex-Soviet republic since coming to power in 2005 on the back of a violent public uprising, or 'tulip revolution.'
Bakiev's rule has been marred by economic problems, high-profile murders, prison riots, and disputes over the control of lucrative businesses.
Marat Sultanov, the speaker of the Kyrgyz parliament, said that the Kyrgyz opposition and pro-presidential forces have come to terms over a new draft Constitution.
"For the sake of national stability, we have reached a consensus and agreed to deal with unsettled issues within the bounds of the law," Sultanov said.
He said a conciliation commission, comprised of opposition and pro-presidential MPs, "have found solutions that suit everyone."
"Tomorrow we will submit a new Constitution to parliament," he said.
Neither the president nor his spokesman have not yet commented on the new deal, which will face a vote in parliament, and need a quorum of 51 of 75 votes to come into force.
During the turmoil in central Bishkek, a television channel broadcasting the rally was taken off the air. The NTS channel had been broadcasting the rally live without comment for six days.
The Russian foreign ministry urged political sides to start constructive dialogue to resolve the conflict.
"We believe that the best way to resolve the current situation can be found only by legal means, through constructive dialogue between government bodies and the opposition," the ministry said in a statement.
"Russia, which wants gradual development in Kyrgyzstan under conditions of stability, hopes common sense will prevail over political emotions and ambitions," the statement said.
A Kyrgyz deputy interior minister denied that Kyrgyz authorities used any weapons against the demonstrators during the clashes, but said that they were ready to do so should there be any attempt to seize government facilities.
