One of Yanukovych's allies said the leader was considering moving up presidential and parliamentary elections and ruled out use of force when trying to break up demonstrations, the Kiev Post reported.
However, members of the president's Party of Regions in parliament Tuesday said Yanukovych has compromised enough by accepting the resignation of Prime Minister Mykola Azarov -- which triggered the resignation of Yanukovych's Cabinet -- repealing stringent anti-protest laws, and accepting conditional amnesty to release anti-government protesters held since the demonstrations began three months ago.
Protests began in November after Yanukovych nixed a long-planned trade and association agreement with the European Union. He later signed an agreement with Russia that included a $15 billion aid package.
Meanwhile, Western support for Ukraine's pro-democracy EuroMaidan movement was growing, with former U.S. President Bill Clinton voicing his support for the opposition, the Post said.
"Kudos to brave Ukrainians demanding real democracy. Urge dialogue & peaceful resolution to achieve a strong, united Ukraine. They can do it!" Clinton posted on his Twitter account.
He is the latest in a long list of politicians and celebrities who have voiced support for the mass anti-government protests.
The endorsement by Clinton, who is popular in the Ukraine, likely will increase pressure on Yanukovych to reach a compromise with the demonstrations, the Post said.
The United States and European Union also were pressuring Yanukovych to reach a settlement with the opposition. EU foreign policy chief Catherine Ashton and U.S. Assistant Secretary of State Victoria Nuland were to visit Kiev this week.