Speaking on the eve of an European Union summit in Salzburg, Austria, where British prime minister Theresa May will brief fellow leaders before they huddle on Thursday to agree on how to play what they hope will be the end-game with London, Mr Barnier called the next summit, in Brussels on October 18, the "moment of truth".
But Tusk said that was now contingent on May reworking her Chequers plan by Ocotber.
May has been urging her European counterparts to accept her controversial Chequers plan, which has divided the Conservative party and the United Kingdom in general after it was thrashed out back in July.
This would keep Northern Ireland under EU economic oversight if London and Brussels can not agree a trade pact to keep UK-EU borders open after a transition period ends in 2020 - an idea that Mrs May and a small party in the province that props up her minority government oppose.
May said she would "shortly" bring forward new proposals on the so-called backstop on how to carry out regulatory checks on goods going in and out of Northern Ireland.
"If the political will is there on the other side I am confident we will reach a deal which is in the EU's interests as well as the UK's", she said.
But despite May's appeal at an EU summit in Austria, European Council President Donald Tusk insisted that parts of her offer are still not satisfactory more than 18 months into the negotiations and must change to keep alive hopes of concluding a Brexit deal in coming weeks.
In a surprise move on Tuesday ahead of the latest European Union summit, the bloc's chief negotiator, Michel Barnier, said Brussels may be "ready to improve" its bargaining position in relation to the 500-kilometer-long border between the Republic of Ireland and the UK. On July 6, 2018, after a marathon 12-hour showdown at Chequers, May finally managed to secure a Brexit plan to present to the EU.
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"If we feel that we are able to finalise and formalise our deal in November, I will call this extraordinary meeting, but not as an emergency but as a punchline of effective negotiations before October and during our October EU council meeting", he said.
He described Brexit as a "a choice pushed by certain people who predicted easy solutions" and claimed that "those who said you can easily do without Europe, that it will all go very well, that it is easy and there will be lots of money, are liars".
On the issue of keeping the Northern Irish border open and "frictionless", to avoid a visible border that would risk reigniting old tensions, May said the UK's proposal was "the only serious and credibly proposition on the table for achieving that objective".
His comments quickly sparked a conversation among some of Brexit's most outspoken critics, including MP Tulip Siddiq, a Labour supporter of the anti-Brexit "Best for Britain" group.
This short-term approach was also hinted at by the Environment Secretary Michael Gove when he said in a TV interview on Sunday that the PM's Chequers plan was the right one for now.
It all suggests a fractious summit in Brussels next month. So we will leave on the 29th of March, 2019, and there will be no second referendum.
She told fellow leaders over a summit dinner Wednesday that "we will of course not agree on every detail", but "the onus is now on all of us to get this deal done".
On Brexit negotiations, she said: The withdrawal agreement is virtually agreed.