"We will not concede in principle on any positions. Russia now understands this. Russia is already beginning to talk constructively," Ukrainian negotiator and presidential adviser Mykhailo Podolyak said in a video posted online.
"I think that we will achieve some results literally in a matter of days," he said.
RIA news agency quoted a Russian delegate, Leonid Slutsky, as saying the talks had made substantial progress.
"According to my personal expectations, this progress may grow in the coming days into a joint position of both delegations, into documents for signing," Slutsky said.
Neither side indicated what the scope of any agreement might be.
Their public comments were issued almost at the same time. They came on day 18 of the war which began when Russian forces invaded Ukraine on February 24 in what the Kremlin terms a special military operation.
Thousands of people have been killed and more than 2.5 million people have fled the fighting. Ukraine has said it is willing to negotiate, but not to surrender or accept any ultimatums.
In a tweet, Podolyak said Russia was carefully listening to Ukraine's proposals. "Our demands are - the end of the war and the withdrawal of (Russian) troops. I see the understanding and there is a dialogue," he said.
Three rounds of talks between the two sides in Belarus, most recently last Monday, had focused mainly on humanitarian issues and led to the limited opening of some corridors for civilians to escape fighting.
Russian President Vladimir Putin said on Friday there had been some "positive shifts" in the talks, but did not elaborate. On Saturday the Kremlin said the discussions between Russian and Ukrainian officials had been continuing "in video format".
