That could force another extension of the date for the United Kingdom's split from the 27 other member states, the third since Britons voted in a June 2016 referendum to quit the EU.
"Hope is the last thing to be lost".
Thursday, October 17-Friday, October 18: EU leaders including Mr Varadkar and Mr Johnson will attend the latest Brussels EU summit.
However, Tusk said in comments broadcast by Polish broadcaster TVN 24 that "certain doubts have appeared from the British side", a reference to Johnson's need to win over politicians who fear he may have conceded too much.
"We don't know the exact details yet, but we now have the possibility of a deal being talked up by Michel Barnier in the last hour".
With massive concessions from the United Kingdom, the European Union leaders can be expected to support the above scenario in the upcoming summit, but Johnson effectively has only six days to formalise the negotiations domestically.
British Prime Minister Boris Johnson has pledged to take the United Kingdom out of the European Union on that day whether or not a withdrawal agreement has been reached.
"There is a pathway to a possible deal but there are many issues that still need to be fully resolved", Varadkar said in a speech on Wednesday.
Reports that Britain has softened its stance on the customs status of Northern Ireland in order to clinch an accord at this week's European summit had raised hopes that a chaotic "no-deal Brexit" can be avoided and driven the pound higher.
"On Brexit the less we say now the better", he added, implying that delicate negotiations are taking place that are better kept under wraps for now.
Britain's Prime Minister Boris Johnson is seen outside the venue for the Conservative Party annual conference in Manchester October 1, 2019.
So I think, as my Taoiseach has said, a deal is possible, and it's possible this month, may even be possible this week.
Difficulties centred on trade and the status of the Irish border, although the EU's chief Brexit negotiator, Michel Barnier, was quoted as saying he was optimistic about a deal.
Johnson has promised to both follow the law and get Britain out by October 31 - a contradiction that might end up being settled in court.
The optimism coincided with Irish broadcaster RTE reporting that two sources confirmed that British negotiators will bring forward an updated plan on Tuesday to deal with the issue of customs and the Irish border.
The sticking point remains the issue of the Northern Ireland backstop meant to guarantee there is no return of a hard border with the Republic.
The two-day European Union summit is crucial because Johnson must get a new deal approved by MPs by Saturday if he is to avoid asking for a Brexit delay. The PM told Tory MPs that the government was "on the Hillary Step" in relation to the Brexit talks, referring to what was famous as the most perilous part of the route up Everest.
The government's parliament representative, Jacob Rees-Mogg, a leading member of the Conservatives' anti-EU lobby, insisted that Johnson has the Brexit credentials to succeed.
He said: "The talks are ongoing".
Johnson postponed a meeting of his political cabinet to Wednesday, when it may become clearer whether a Brexit deal will be done this week, and the government will then be able to decide whether to call MPs in for a sitting Saturday.