Arseniy Yatsenyuk, who heads the Fatherland Party, would become the Prime Minister, and if he agrees the statement said, Yanukovych will dismiss the current government, which has been one of the protesters' demands.
The President also offered the position of Deputy Prime Minister on humanitarian issues to boxer Vitaly Klitschko, leader of the Udar Party, and agreed to a working group to look at changes to the constitution, according to the statement.
Neither Yatsenyuk nor Klitschko have agreed to the offer, which comes after a day of tension between police and hundreds of protesters who continue their vigil in Kiev's Independence Square and surrounding streets.
They have set up tent cities and put up barricades.
Large plumes of black smoke billowed from the barricades Saturday as riot police watched and waited nearby.
Protests against the government began in November when Yanukovych refused to sign a trade agreement with the European Union in favor of stronger economic ties with Russia.
The opposition party UDAR warned ealier Saturday that authorities are preparing to provoke peaceful protesters and may plant aggressors among the crowds in an effort to enact emergency rule.
"We call on all Ukrainian citizens not to succumb to provocations of authorities and continue the protest exclusively in a peaceful manner," a party statement said.
A group of protesters blocked the entrance to the Energy Ministry on Saturday, a move Ukraine's Energy Minister Eduard Stavytsky called a direct threat to the entire energy system.
Some protesters entered the building and argued with Stavytsky.
"Your action is illegal," Stavytsky told them.
"You have crossed the red line. You have entered a high security building."
The protesters left the building saying they did not intend to undermine the work of the ministry in any way.
The European Union is getting involved in the crisis.
Next week EU's foreign policy chief Catherine Ashton and a European Parliament delegation are scheduled to visit Ukraine.
Ukraine's richest oligarch Rinat Akhmetov issued a statement Saturday saying business cannot keep silent when people are killed.
"A real danger of breakup of the country emerges when a political crisis can lead to a deep economic recession, and thus inevitably result in lower standards of living," he said.
Police say one officer was killed and another one stabbed in clashes with protesters.