Barclay and Barnier's talks come ahead of a crunch EU summit on October 17 and 18, which is seen as the last chance for the UK and the EU to agree a deal ahead of the Brexit deadline, the BBC reported.
The UK is due to leave the EU at 11 p.m. (local time) on October 31.
A day after EU leaders accused the UK of proposing untested ideas, Johnson and Varadkar held "detailed and constructive" talks for more than two hours on Thursday, with the latter saying the "very positive" meeting meant negotiations could resume in Brussels.
The UK cabinet was briefed on the latest developments on Friday morning.
Education Secretary Gavin Williamson told the BBC that it had been a "very nice" conversation, but added that he would not provide a "running commentary" on events.
With three weeks to go until the Brexit deadline, Johnson continues to insist the UK will leave the EU with or without a deal at the end of the month.
That is despite the so-called Benn Act - passed by MPs last month - demanding he request a delay to the Article 50 deadline from the EU until January 2020 if a deal has not been agreed before October 19.
Should any new deal be reached between the EU and UK, it will still have to be approved by MPs in Parliament.
( With inputs from IANS )