He said they had had “serious detailed discussions” and things were “moving forward with momentum”.
Mr Barnier said it had been a “cordial” meeting, but “lots of work has to be done in the next few days”.
The deadline for the UK to exit the EU is 31 October.
On Thursday, European Commission President Jean-Claude Juncker said a new Brexit deal could still be reached by then.
Prime Minister Boris Johnson said some “progress” was being made, although it was important not to “exaggerate” this.
But Irish Foreign Minister Simon Coveney told the BBC on Friday that there was a “wide gap” between the UK and the EU, with Brussels “still waiting for serious proposals” from London.
The backstop – the policy aimed at preventing the return of a hard border on the island of Ireland after Brexit – has proved the biggest point of contention in EU-UK talks so far,
It was a major sticking point in former Prime Minister Theresa May’s attempts to get Parliament to back her withdrawal agreement, which was rejected three times by MPs.