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One of the president's main foes described his offer as a "poisoned" attempt to divide the opposition and kill off mass protests. The demonstrations erupted late last year when Yanukovich ditched landmark agreements with the European Union and opted instead for closer ties with Russia.
Emboldened opposition leaders said they would press for more concessions, including early elections, setting the stage for a tough political battle when parliament meets for a special session on Tuesday.
The two-month standoff has sparked the worst violence in Ukraine since it won independence in 1991, as the Soviet Union collapsed. At least six people have been killed, according to the prosecutor's office and medics, and the crisis has deepened tension between Russia and the West.
For the opposition, accepting Yanukovich's offer to serve under him in a revamped government carries the risk of breaking faith with thousands of peaceful demonstrators, as well as alienating more radical protesters over whom it has only tenuous control.
"Yanukovich's offer always appeared as a poison chalice for the opposition - meant to divide the opposition, and boost his chances in the March 2015 presidential election," Tim Ash of Standard Bank said.
In the latest violence on Sunday, a few thousand protesters tried to storm an ornate cultural centre where hundreds of security forces personnel were gathered in central Kiev, a few hundred metres from the hub of weeks of opposition protests on Independence Square.
In a two-hour pre-dawn confrontation, demonstrators threw stones and smoke bombs while police fired stun grenades and sprayed water into the crowd.
Police and security forces later left the building, its windows shattered, and streamed out through a corridor created by the crowd after an opposition leader, Vitaly Klitschko, arrived at the scene and helped to negotiate a solution.
The opposition planned to hold a prayer ceremony later on Sunday for the protesters who have been killed. A coffin bearing the body of one of them, Mykhailo Zhyznevsky, was borne through the streets of Kiev before his burial, with several hundred people marching behind.
Zhyznevsky, a Belarussian living in Ukraine, was one of three people officially recognised by the prosecutor's office as having died from gunshot wounds after clashes last week. He would have been 26 on Sunday,
Yanukovich abruptly abandoned plans to sign political association and free trade deals with the EU in November, pledging instead to improve ties with former Soviet master Russia and angering millions who dream of a European future.
The unrest has spilled over into other regions of the country of 46 million people. Protesters have occupied municipal headquarters in up to 10 places, many of them in western Ukraine where opposition to Yanukovich's rule is strongest.
Hoping to end protests that threaten to bring the country to a standstill, Yanukovich on Saturday offered former economy minister Arseny Yatsenyuk the post of prime minister.
Klitschko, a former international boxing champion, was offered the job of deputy prime minister responsible for humanitarian issues, the presidential website said.
The presidency linked its offer to the opposition reining in violent protesters. Though the protest movement is largely peaceful, a hard core of radicals have been fighting pitched battles with police away from Independence Square.
Opposition leaders said they would press their calls for early elections and repeal of an anti-protest law.
"We are ready to take on this responsibility and take the country into the European Union," Yatsenyuk was quoted as telling crowds on Independence Square after emerging from talks with Yanukovich. But he added that this would entail the freeing of former prime minister Yulia Tymoshenko, the president's arch-opponent, who was jailed in 2011.
Klitschko told German newspaper Bild am Sonntag: "This was a poisoned offer by Yanukovich to divide our protest movement. We will keep on negotiating and continue to demand early elections. The protest by Ukrainians against the corrupt president must not have been in vain."
Opposition leaders say Yanukovich has betrayed Ukraine and are calling for an election long before the next one is due in spring 2015. Klitschko said it must be held this year.
The United States has warned Yanukovich that failure to ease the standoff could have "consequences" for its relationship with Ukraine. Germany, France and other Western governments have also urged him to talk to the opposition.
"The situation is tense and serious. The days ahead will determine which way Ukraine heads in the future. I believe that there is only one piece of advice we can give: Ukraine's path into the future cannot be found with the use of violence, no matter which side it comes from," German Foreign Minister Frank-Walter Steinmeier told reporters on Sunday.
Pope Francis called for an end to the violence and for all the parties to engage in "constructive dialogue".
Russia on Saturday stepped up its warnings against international interference in Ukraine, telling European Union officials to prevent outside meddling and cautioning the United States against inflammatory statements. President Vladimir Putin is due to visit Brussels on Tuesday for what promises to be a tense EU-Russia summit.