Mr Barnier warned that the Prime Minister would be able to send a modified Irish offer in Brussels if she wanted to conclude a withdrawal deal this month.
In response to forthcoming business reports, he declined to enter into speculation about the possible dates of the UK-EU agreement.
Speaking at the Finnish Prime Minister Juha Sipil's meeting in Helsinki, Barnier said: "Is the deal possible before next week? Clock is being marked, I will not give you a date. I can not give you a date.
"Issues that are crucial and clear, fair, and responsible for the quality, sustainability and credibility of Ireland and Northern Ireland remain crucial issues that are decisive.
"There is more work to do, I'm determined."
Mr Barnier is in the capital of Finland at the European People's Party summit with Leo Varadkaru, who is still important in Brex's talks as Irish Prime Minister.
Mr Varadkar believes that the option of the November agreement is dying every day, without success in the talks.
He told journalists the hopes for Brex's November agreement, which is "less likely to happen on a daily basis."
The Brex talks have concluded the so-called "tunnel", a private negotiating country, again because British and EU officials are continuing to negotiate in Brussels.
The 27 ambassadors' meeting will be held on Wednesday evening, which was postponed to Friday.
The changes took place after the May Cabinet was given an indication to begin preparations for this week's meeting in order to complete the United Kingdom's forthcoming offer of the Irish brochure before it was delivered to Brussels.
The Irish border remains one of the obstacles to the successful conclusion of Brexi withdrawal deals.
Ms. Maiya also told her PM that she would not agree "free of charge" and that any deal with Brussels would depend on a "acceptable" system for future relations with the EU.
According to sources, the deal could be signed and delivered by the prime minister in Brussels on Friday.
One EU diplomat told Express.co.uk that it would restore plans for the November summit, which could take place as soon as possible.
The possible transaction dates are "22, 23, 24 November," according to a senior official from another EU negotiator.
However, Mrs Maiya wants Parliament to bring Brexit deal before Christmas, thus signing an agreement in November.