"If we really want to preserve peace and agreement, and really want to build a legitimate democratic society that we so often talk about ... then let's hold new elections," the Russian news agency on Monday quoted Kuchma as saying.
Earlier, Prime Minister Viktor Yanukovich said he would agree to stage a new presidential vote in two regions if mass fraud were proven to have occurred in the 21 November election.
"If there is proof of cheating, that something illegal occurred there and if there is no doubt among experts, I will agree with such a decision," he said in televised comments on Monday, referring to two regions in his native eastern Ukraine.
Yanukovich was declared the winner in the run-off election but liberal candidate Viktor Yushchenko has challenged the outcome in the Supreme Court, citing widespread fraud.
Planned new vote
Yanukovich was shown on television telling Kuchma and aides that he was prepared to go along with a plan for a new vote in two of his eastern strongholds - the regions of Donetsk and Lugansk - where Yushchenko supporters have alleged massive voting fraud took place.
The results of the poll has been
bitterly disputed on the streets
Ukraine's top court on Monday examined the allegations as the country's civilian leadership scrambled to quell talk of national disintegration.
As the political crisis that has rocked the 48-million-strong eastern European state for the past eight days continued, Kuchma warned that the country's financial system "could fall down like a house of cards" within days if the dispute were not resolved.
And with one pro-Russian region in east Ukraine pushing ahead with plans to establish greater autonomy from Kiev, Defence Minister Oleksandr Kuzmuk cautioned the armed forces would uphold Ukraine's stability, sovereignty and territorial integrity "no matter what the circumstances".
The regional parliament of Ukraine's pro-Russia Donetsk province meanwhile decided to hold an emergency session on Tuesday over a regional referendum on autonomy scheduled for the weekend, a local administration spokeswoman said on Monday.
"If our people want autonomy no one can tell us we can't have it," Olena Bondarenko said.
Such talk in Donetsk and some parts of eastern Ukraine has triggered alarm bells in the West as well as in other areas of eastern Ukraine and in Moscow where officials have been working furiously to quell any breakaway moves.