Vitali Klitschko said he would lead pro-EU protesters “on the attack” in the capital, Kiev, if the government refused to call snap elections.
PM Mykola Azarov said compromises “might be possible”, but the opposition should avoid ultimatums.
Two activists were killed in clashes with police in Kiev on Wednesday.
Prosecutors confirmed they had died from gunshot wounds.
They were the first fatalities since the anti-government protests flared up in late November over Viktor Yanukovych’s decision to pull out of a landmark treaty with the EU.
Late on Wednesday, Ukraine’s Radio Liberty reported the death of a third activist.
The body of Yuri Verbitsky was found in a forest outside Kiev, bearing signs of torture, according to the broadcaster.
He had reportedly been abducted earlier this week with activist Igor Lutsenko, who was later released. Igor Lutsenko is said to be in hospital.
Hundreds of people have been injured in the clashes, though some of the violence has been blamed on a little-known far-right group, Right Sector.
Wednesday’s unrest came on the day that new anti-protest laws entered into force. Parliament approved the laws last week, triggering renewed protests which spilled into violence on Sunday night.
As dawn broke on Thursday, the barricades were still burning, billowing black smoke from the piles of tyres that now mark the front line between the riot police and the protestors.
Speaking at a mass rally on Wednesday evening, Vitali Klitschko said the president could end the stand-off “without bloodshed” by calling early elections, but that “tomorrow, if the president does not respond… then we will go on the attack”, to roars of approval from the crowd.
Vitali Klitschko said police were preparing to clear demonstrators out of the main protest encampment at Maidan (Independence Square).
“We must do all we can to stop them clearing us out,” he told demonstrators.
Another opposition leader, Arseniy Yatseniuk, said the government had 24 hours to respond to the demands, which also include the lifting of the new anti-protest laws.
“If this does not happen, we will march forward together. If it’s a bullet to the head, then it’s a bullet to the head,” he declared.
But the prime minister said opposition leaders should be “more humble”.
“The opposition leaders should move away from the language of ultimatums,” Mykola Azarov said.
“We are ready to compromise, to agree. The opposition leaders should understand that they also bear responsibility in avoiding a civil war, and bloodshed, and so does the government.”
Wednesday’s violence began in a small area around Hrushevskyy Street, a road leading to government buildings close to the protest camp at Maidan.
Security forces later fell back to their positions after fierce clashes with protesters, but by the afternoon had pushed on through the barricades.
Protesters again hurled petrol bombs and stones while riot police responded with stun grenades and rubber bullets.
Thousands of protesters also gathered in Independence Square.
There was a crush at one of the narrow entrances into the square when protesters trying to get in met protesters who were trying to get out to fight the police.
At least two ambulances were seen carrying away the wounded.
Officials confirmed two bodies were found with bullet wounds close to the scene of the clashes.
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