EU and UK negotiators will enter a fourth and last scheduled round of talks this week that could determine if a comprehensive new agreement is struck by the year-end deadline, AFP reported.
Britain formally left the other 27 EU nations in January but still largely operates as if it were a member of the bloc.
It also continues making contributions to the EU budget – a reality that particularly upsets Brexit supporters.
British Prime Minister Boris Johnson has vowed not to extend the talks past the current deadline – something he must do by the end of June – and the prospects of a broad new deal look bleak.
At a London seminar on Friday, Barnier’s senior adviser, Stefaan de Rynck, revealed he thought there was little chance of a deal being agreed by the 30 June deadline.
Britain has accused the EU of wanting to string out the talks until the November deadline for an agreement in the hope of making the UK cave in to its demands.
With the latest round of negotiations beginning on Tuesday, Johnson wants to up the pace of the talks after making it clear that he will not extend the transition period and will not budge on sovereignty issues such as fishing rights.
Ministers are anxious to ensure that businesses have as much time as possible to prepare for whatever trading regime is in place when the UK’s current arrangements come to an end on December 31.
Their aim is to make significant progress before Johnson holds a meeting with European Commission President Ursula von der Leyen in July, at which the prime minister will formally reject an offer of an extension.
But there are fears that EU leaders think that by running down the clock Britain will fold at the last minute. Johnson has made clear he will not blink and is prepared to leave the EU without a trade deal if necessary.
Barnier told The Sunday Times that London and Brussels could afford to make the economic situation even worse by breaking off their nearly 50-year partnership without arrangements for what comes next.
"If we don’t get an agreement then that will have even more consequences. And then of course those will be added to the already very serious consequences of the coronavirus crisis," Barnier said.
"So I think that we have a joint responsibility in this very serious crisis, which affects so many families... with so many deaths, so many people sick, so many people unemployed... to do everything we can to reach an agreement and I very much hope that we will do so."
The previous round of talks ended in acrimony in May.
The European Union is willing to offer Britain preferential trade terms if Johnson signs up to the major standards and regulations followed by the remaining members of the bloc.
Johnson's team argues that the whole point of Brexit was to give Britain the right to set its own rules.
Barnier accused UK negotiators of reneging on the commitments Johnson signed up to in a non-binding political declaration that accompanied the sides' formal divorce deal.
"The UK has been taking a step back – two steps back, three steps back – from the original commitments," Barnier told The Sunday Times.
"The UK negotiators need to be fully in line with what the prime minister signed up to with us."