Barnier was expected to brief the European Parliament's Brexit Steering group on progress on Friday afternoon.
Meanwhile, European Council president Donald Tusk handed the PM a last-minute reprieve to secure an agreement, but warned the United Kingdom still has not presented a "workable, realistic proposal".
The surge in the pound on Friday pushed it back above $1.26 for the first time since Boris Johnson became leader of the Conservative party.
Johnson said late Thursday there was a "pathway" to a belated agreement to stave off a chaotic, costly no-deal Brexit on October 31, while Varadkar said the meeting was "very positive".More news: DISCOVERY | 20 new moons puts Saturn ahead of Jupiter
In a statement on Friday, Arlene Foster said that, while negotiations are a matter for the UK Government, the DUP will "exercise our considerable influence" to "stand up for Northern Ireland" and will only ever support arrangements that are in province's long-term economic and constitutional interests. "You need vigilance, determination and patience", following his meeting with the UK's Brexit Secretary Stephen Barclay.
This would be a major breakthrough and a cause for celebration in Ireland and certain parts of the UK- already the pound has risen significantly against the dollar - but Mr Johnson is set for huge backlash from the DUP if has indeed agreed for Northern Ireland to be treated differently to the rest of the UK.
The tunnel is European Union jargon for line-by-line negotiations held by a small team of officials from the United Kingdom and European Union which are conducted in complete secrecy, meaning that offers and concessions are not disclosed.
He nevertheless warned that there was "no guarantee of success" and said that time was "practically up".
The question of the Irish border - and the proposed "backstop" that angers Brexiteers - has become the most intractable issue in negotiations.More news: How to get new, more detailed spoken instructions from Google Maps
France's European affairs minister, Amelie de Montchalin, said early Friday that a no-deal Brexit "is probable, at this stage". If it is time for the sake of time? "Barnier will have to say whether we can or can't start negotiating a text", a European source told AFP.
The EU's chief negotiator Michel Barnier is now meeting with Britain's Brexit minister Stephen Barclay in Brussels.
A British spokesman also described the closed-door talks as a "constructive meeting".
Johnson's Brexit envoy, Stephen Barclay, drove into European Union headquarters for a Friday breakfast meeting with Michel Barnier at which he is expected to brief the European Union negotiator on what, if any, fundamental breakthrough has been made.
The tepid positivity came after Mr Johnson and Mr Varadkar said they could "see a pathway" to a possible agreement following face-to-face discussions in the Wirral.More news: Things WWE Fans Need To Know About Tyson Fury
The main stumbling block remains how the U.K.'s only land border, between Britain and Ireland, is dealt with.