He said a cross-party commission would be set up on Monday to try to resolve the deepening crisis. Opposition leaders confirmed this.
Earlier, a group of protesters - trying to reach parliament - clashed with police. Dozens of people were injured.
The US and EU called for an end to the violence and urgent political talks.
The violence broke out as many thousands of protesters held a rally in Kiev's Independence Square, outraged by new laws which they said restricted basic freedoms.
The ruling party of Mr Yanukovych denies this, saying the legislature is in line with European standards.
However, Western countries have expressed deep concern at the new laws.
The anti-government movement began in protest at Mr Yanukovych's decision in late November to pull out of a landmark treaty with the EU, but has expanded to demand his resignation.
'Civil war' warning
Late on Sunday, President Yanukovych's press office said a "working group" headed by National Security and Defence Secretary Andriy Kluyev would be set up.
It said the group - made of members of government and the presidential administration - would meet opposition representatives on Monday to try resolve the crisis.
Opposition leader Vitali Klitschko confirmed this after meeting President Yanukovych at his residence outside Kiev.
"We must use every opportunity to resolve the crisis peacefully," the former world heavyweight boxing champion, who leads the Udar party, told Ukraine's Hromadske TV.
He warned against a "scenario of force", adding that he "didn't rule out the possibility of a civil war".
Arseniy Yatseniuk, another opposition leader, said Mr Yanukovych personally called him to say that he was ready for talks.
However, the opposition warned that the talks must produce real results and not be an opportunity for the president to play for time. The opposition is demanding the resignation of the government and snap presidential elections.
But opposition leaders are under huge pressure to come up with an action plan, amid criticism from many activists that their campaign has been too passive.
'Provocateurs and extremists'
Sunday's rally in Kiev, attended by many thousands, began with calls from opposition politicians to disregard the new laws curbing protests that pro-EU demonstrators have been staging for the past two months.
But clashes erupted as a group of demonstrators - reportedly far-right activists - headed away from the main square towards parliament to express their outrage.
They ran into police and interior troops cordons near the Dynamo football stadium, some 300 metres from Independence Square.
The demonstrators pelted police with flares, thunder flashes and petrol bombs, the BBC's Daniel Sandford in Kiev reports.
They overturned a bus used by police and set it alight. Other vehicles were also set on fire.
Police - who were sheltering behind buses under their riot shields - threw their own thunder flashes and gas canisters to try to force the crowd back.
Hours after darkness fell, the stand-off was continuing on Hrushevskyy Street, some 100 metres from Independence Square.
Police also used water cannon to try to disperse the demonstrators.
Interior ministry spokesman Serhiy Burlakov blamed "provocateurs and extremists" for the confrontations and urged people not to follow their lead.
Police were filming everything and had opened criminal proceedings under Article 294 (organisation of mass riots), the interior ministry said.
The opposition leaders said they were committed to a peaceful resolution of the crisis, denouncing those activists who took part in clashes.
'Enslaving' laws
During the day, the rally on the main square heard a call from a former Ukrainian navy chief for members of the armed forces to defy "illegal" orders from those in power, Unian news agency reported.
"Tomorrow the regime will enslave you too. Therefore we are calling on you to fulfil your military oath of loyalty to the Ukrainian people and not to the authorities who have gone off the rails," said Rear Admiral IhorTenyukh, who was sacked by Mr Yanukovych in 2010.
By BBC
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