Supporters of Viktor Yanukovych hold a rally near the Central Election Commission for a second day
Supporters of defeated Ukrainian election candidate Yulia Tymoshenko say they want to challenge the results of Sunday's vote in court.
An MP from her electoral bloc said that it suspected vote-rigging in 1,000 polling stations.
Mrs Tymoshenko has reportedly told allies she will "never" accept Viktor Yanukovych as the winner of the poll.
However, with nearly 100% of the votes counted, foreign monitors are saying the election was free and fair.
Observers from the Organisation for Security and Co-operation in Europe (OSCE) were unusually blunt, warning the country's political leaders they should listen to the people's verdict and make sure the transition of power was peaceful, the BBC's Richard Galpin reports from the Ukrainian capital, Kiev.
We will do everything to prove that this election was falsified
On Tuesday, Russian President Dmitry Medvedev congratulated Mr Yanukovych on the completion of "an election campaign highly rated by international observers and on the success achieved in the presidential poll," the Kremlin said in a statement.
With 99.94% of votes counted after Sunday's poll, Mr Yanukovych had won 48.94% to 45.48% for Mrs Tymoshenko.
Yanukovych supporters have been gathering in Kiev in anticipation of the announcement of the full election results, which are expected later on Tuesday, according to Russia's Interfax news agency.
His discredited victory in the 2004 presidential election, which was marred by widespread fraud, sparked the Orange Revolution, led by Mrs Tymoshenko's former ally, Viktor Yushchenko.
The latter was elected president when the result was annulled and the election re-run, but his popularity quickly declined and he was eliminated from the first round of this year's election.
Ukraine faces serious economic problems as well as internal divisions over whether to look to the EU and Nato, or Russia.
Speaking at a news conference, Tymoshenko MP Andriy Shkil said the results from 1,000 polling stations could be challenged in court.
Ukraine is still awaiting the final results of the election
"We will recognise the victory only if we fail to prove the violations that resulted in this victory, in the courts," said Mr Shkil.
But the final decision to mount a challenge will be taken by Yulia Tymoshenko herself and she has not spoken in public since Sunday night when she denounced Mr Yanukovych for declaring victory so quickly after voting ended, our correspondent says.
She scheduled news conferences in Kiev twice on Monday but then cancelled them. Her spokeswoman told the BBC Mrs Tymoshenko would speak on Tuesday but later reports suggested she might issue a statement instead.
In a statement published on the bloc's website, deputy leader Olena Shustik said the vote in eastern and southern Ukraine had been "distorted because of fraud".
Ukraine's Ukrainska Pravda daily quoted Mrs Tymoshenko as telling her party at a private meeting on Monday: "I will never recognise the legitimacy of Yanukovych's victory with such elections."
She had, the paper said, instructed her lawyers to prepare to contest the electoral results in court.
However, the paper added that some members of her bloc disagreed with her, calling instead for her to acknowledge defeat, step down as prime minister and move into opposition.
Praise for ballot
International monitors described the election as an impressive display of democracy.
The OSCE's head of mission, Joao Soares, said Ukraine's electoral commission had been transparent and unbiased.
The reign of the Orange movement was a complete failure for the country
The EU and US both welcomed the conduct of the election, without referring to Mr Yanukovych as the winner. Mr Yanukovych is seen by analysts as being closer to Moscow than Brussels.
"I should in particular like to congratulate the people of Ukraine for the high turn out in both rounds of the elections and the strong commitment demonstrated to the democratic process," said EU foreign affairs chief Baroness Ashton in a statement.
The EU, she added, remained "committed to deepening the relationship with Ukraine and supporting it in implementing its reform agenda" and looked "forward to working with the new president to this end".
The US embassy in Kiev described the election as "another step in the consolidation of Ukraine's democracy".
The Russian ambassador to Ukraine, Mikhail Zurabov, said he expected Russian-Ukrainian relations to improve whoever assumed the presidency.
"One can mention dozens of projects which could long since have been implemented for the benefit of the two countries, but which either have not received any attention or have been stopped over the past five years," he said, referring to Mr Yushchenko's tenure.
Mr Zurabov added that he now believed a "reset" would take place.
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