Thousands of Yushchenko supporters are continuing street protests against Prime Minister Viktor Yanukovych.
The dispute over his 21 November poll victory threatens to split Ukraine.
The BBC's James Coomarasamy in the capital, Kiev, says the pressure seems to be mounting on Mr Yanukovych, widely seen as an ally of Russia.
The head of his electoral campaign has resigned and the outgoing President Leonid Kuchma, who backed Mr Yanukovych, has joined calls for fresh elections.
Parliament's no-confidence motion would have no legal effect, but correspondents say that if it were passed, it would be a big symbolic victory for Mr Yushchenko, a pro-Western liberal.
Mr Yanukovych said he would agree to a partial re-run of the poll if there was evidence of mass violations.
Heated negotiations
The dispute has led to more than a week of mass protests by supporters of both Mr Yushchenko and Mr Yanukovych.
Mr Yushchenko accused his rival's negotiators of "sabotaging" talks aimed at resolving the crisis, and threatened to withdraw from them.
On Tuesday he rejected an offer by Mr Yanukovych to appoint him prime minister or for both men to withdraw from new elections, the AFP news agency reported.
Mr Yanukovych's backers in eastern and southern regions of the country have threatened to demand autonomy if Mr Yushchenko becomes the new president of Ukraine.
But parliament in the Donetsk coal-mining region has now postponed an emergency session to discuss a referendum on autonomy. The region's governor has been called to Kiev for urgent talks.
In Moscow, an ally of President Vladimir Putin, parliamentary speaker Boris Gryzlov, warned that the situation in Ukraine "is heading towards a split or towards bloodshed".
The US and the European Union - who have refused to recognise the official results - have expressed concern about the possibility of Ukraine breaking up.
Mr Kuchma warned on Monday that the country's financial system faced collapse after reports from Ukraine's central bank that there had been a rush to withdraw bank deposits.
