The EU's chief negotiator said "genuine progress" and a decision on whether to extend the transition period were both needed by June.
The UK said "limited progress" had been made and talks needed to "move forward in a constructive fashion".
The two sides will hold two further rounds of talks before the end of the transition period in December.
Mr Barnier said a joint decision would be taken on 30 June about whether to extend the transition period.
But the UK government has already said it will refuse to extend it beyond December, even if the EU requested a delay.
Following the talks - which took place using video-conferencing technology because of the coronavirus pandemic - Mr Barnier said: "The UK has affirmed once again this week its wish to make tangible progress between now and June and we're on the same wavelength on this and we respect the same timetable.
A UK spokesperson said: "We do not recognise the suggestion that we have not engaged seriously with the EU in any area.
"We have just had a negotiating round lasting most of a week, including two full days talking about fisheries and three full days discussing so-called level-playing-field issues."
The spokesperson said they were "ready to keep talking" but some of the EU's proposals were "unprecedented" and did not account for the UK as "an independent state".
In a statement, the UK government said it had been a "full and constructive negotiating round".
"However, limited progress was made in bridging the gaps between us and the EU," it said.
"Our assessment is that there was some promising convergence in the core areas of a free trade agreement, for example on goods and services trade, and related issues such as energy, transport, and civil nuclear cooperation."
But it said that the EU's offer on goods trade "falls well short of recent precedent in FTAs (free trade agreements) it has agreed with other sovereign countries".
"This considerably reduces the practical value of the zero tariff zero quota aspiration we both share," the statement said.
The UK government also highlighted "significant differences of principle" in areas including the level playing field and fisheries.
It said talks needed to "move forward in a constructive fashion" and the UK "remains committed to a deal with a free trade agreement at its core".
The next round of talks are due to be held during the weeks beginning 11 May and 1 June.
A truly rich man is one whose children run into his arms when his hands are empty.