Opposition leader Kurmanbek Bakiyev, who has taken the nation's helm after protesters seized government headquarters, says he plans to run.
"I think I should run in the presidential elections. God willing, I will," Bakiyev, who was appointed acting president and prime minister after the lightning-fast changes of Thursday, said.
Akayev's whereabouts were unclear on Saturday, although the Russian news agency Interfax cited unspecified sources as saying he had gone to Moscow late on Friday.
Bakiyev also said he had information that Akayev was in Russia.
The Kremlin press service, the Russian Foreign Ministry and the Kyrgyz embassy in Moscow all declined comment on the reports that Akayev was in Russia.
It was not immediately clear whether elections would also be called for parliament, but former opposition leaders now in power have suggested a new legislative vote would be held some time after a presidential election.
Protests paved the way for the
Akayev government's ouster
A semblance of calm returned to the capital, Bishkek, on Saturday after two nights of looting and sporadic gunfire that marred celebrations of the sudden political change and underline the challenges facing the Central Asian country's new leaders.
Iskander Sharshiyev, an opposition leader whose group has been working with police to restore order, said three people he described as "pillagers" were killed overnight amid clashes between police and looters.
However, Interior Ministry spokesman Nurdin Jangarayev said: "Everything was normal last night - better than the previous night. We were working with volunteers all night. We have calmed the people down."
Akayev apparently has not resigned and the technical legitimacy of the new leaders in power in Kyrgyzstan remains unclear.
Results of disputed February and March parliamentary elections, which triggered the opposition push for Akayev's ouster, were annulled by the Supreme Court after Thursday's takeover.
"I think I should run in the presidential elections"
opposition leader and acting president
But the fallout from those elections was still being felt on Saturday, with two competing national legislatures jockeying for authority.
In the parliament building, members of the restored legislature made up of lawmakers who served before the disputed elections met in one room, while politicians elected in the recent voting gathered in another.
Winners in the voting - which the opposition now in charge said was marred by fraud - challenged the authority of the restored legislature.
"Our opinion is that we should be the legitimate lawmakers, because the people have chosen us," Roman Shin, elected in the recent vote, said.
Shin said the former lawmakers who have returned to parliament "don't want to abandon power".
He said the recently elected legislature has more support than the restored old parliament or the opposition leaders now leading the country.
Thousands rallied against alleged
fraud in polls earlier this year
"The revolution was made by 5000 people," Shin said, referring to the swelling crowd that gathered outside the presidential and government headquarters on Thursday before some of its members stormed the building.
The lawmakers from the restored previous parliament's upper house elected a speaker.
They also discussed the possibility of legitimising interim leader Bakiyev's position by making him prime minister, instead of just acting prime minister.
After a swelling crowd in Bishkek marched to the central square, and hundreds stormed the government headquarters, overcoming riot police who put up little resistance.
"An unconstitutional coup d'etat has been staged in Kyrgyzstan," a statement purportedly from Akayev said.
"My current stay outside the country is temporary. Rumours of my resignation are deliberate, malicious lies."
"An unconstitutional coup d'etat has been staged in Kyrgyzstan"
Ousted president Askar Akayev
In the emailed statement, with the sender listed as the Kyrgyz presidential press service, Akayev said he had given orders not to use force during the uprising, ignoring the advice of his aides, and that he had left the country to avoid bloodshed.
Call from PutinAkayev's departure made Kyrgyzstan the third former Soviet republic in the past 18 months - after Georgia and Ukraine - to see protests bring down long-entrenched leaders widely accused of corruption.
The 60-year-old Akayev had led Kyrgyzstan since 1990, before it gained independence in the Soviet collapse.
Bakiyev said on Saturday that Russian President Vladimir Putin called him on Friday night, and asked in an "amicable conversation" how Russia could help the people of Kyrgyzstan.