Mr Yatsenyuk said the opposition was generally ready to accept leadership, but several key demands must be met, including new elections.
Clashes continued overnight. Activists stormed a Kiev building housing police.
The president’s proposal came amid new efforts to end the deadly unrest.
He offered the post of prime minister to Mr Yatsenyuk and the position of deputy PM to former boxer Vitali Klitschko following talks on Saturday.
Speaking to large crowds in central of Kiev late on Saturday, the opposition leaders repeated their demands.
“Viktor Yanukovych announced that the government wasn’t ready to take the responsibility for the country and offered to the opposition to lead the government,” said Mr Yatsenyuk.
“What is our response to this? We are not afraid of the responsibility for the destiny of Ukraine.”
Later, in a tweet Mr Yatsenyuk said: “No deal @ua_yanukovych, we’re finishing what we started. The people decide our leaders, not you.”
Mr Yatsenyuk says that Tuesday, when a special session of parliament has been called, will be “judgement day”.
The opposition is demanding that a free trade agreement with the European Union be signed and political prisoners be freed, including former Prime Minister Yulia Tymoshenko.
They are also demanding early presidential elections. A vote is not due until 2015.
Mr Klitschko told the crowd that they would press ahead with their demands and that talks would continue.
“We are not turning back and we will keep discussing and trying to find a direction,” he said.
The demonstrations began in November after Ukraine decided not to sign an accord on more co-operation with the EU. Instead, the government opted to deepen ties with neighbouring Russia.
Late on Saturday, hundred of protesters surrounded Ukrainian House, a cultural centre in Kiev defended by riot police. Petrol bombs and fireworks were thrown inside.
But the officers were able to leave the building early on Sunday. Mr Klitscho had negotiated their safe passage, local media reported.
Mr Yatsenyuk is the parliamentary leader of the country’s second biggest party, Fatherland, and an ally of Ms Tymoshenko. Mr Klitschko is the leader of the Udar (Punch) party.
The crisis in Ukraine escalated this week when two activists were killed, and another was found dead with torture marks in a forest near the capital.
A fourth, 45-year-old protester is said to have died in a Kiev hospital on Saturday from injuries sustained in earlier violence.
Although the protest movement – the EuroMaidan – is largely peaceful, a hardcore of radicals have been fighting pitched battles with police away from the main protest camp in Maidan, or Independence Square.
On Friday protesters seized a number of government buildings in Ukrainian cities outside Kiev, particularly in the west, which has traditionally favoured closer ties with Europe, including in the cities of Chernivtsi, Ivano-Frankivsk, Lutsk and Lviv.
On Saturday the protests spread to cities further east, including Vinnytsya, just west of Kiev.