His remarks followed reports that the EU is poised to revise his instructions in the negotiations to allow more leeway to come to a compromise with Theresa May.
Sterling rose by about one per cent against the US dollar and half a per cent against the euro after he spoke.
But the Prime Minister was warned that any draft deal based on her so-called “Chequers proposal” could trigger a “catastrophic split” in Tory ranks.
Mr Barnier delivered his latest assessment of the chances of a deal during a question-and-answer session at a conference in Bled, Slovenia.
He said: “I think that if we are realistic we are able to reach an agreement on the first stage of the negotiation, which is the Brexit treaty, within six or eight weeks.
“Taking into account the time necessary for the ratification process, the House of Commons on one side, the European Parliament and the Council on the other side, we must reach an agreement before the beginning of November.
“I think it is possible.”
Following his remarks, Downing Street acknowledged that there had been a welcome acceleration in the pace of the Brexit negotiations.
The Prime Minister’s spokesman said: “We’ve said ourselves that we are focused on securing a deal in October and that continues to be what we’re working towards.
“We’ve had positive engagement over the summer with EU leaders in relation to the Chequers plan. That engagement obviously continues.”
He added: “You’ve seen over the summer an intensification in the talks – that’s something we’ve called for. You’ve seen progress continuing to be made in relation to that withdrawal agreement.”
British and EU diplomats hope an informal summit of EU leaders in Salzburg, Austria, next week could turn into a significant moment in the Brexit talks.
Ambassadors of EU nations in Brussels are understood to have been told the gathering is likely to discuss giving Mr Barnier additional guidelines for his approach to the negotiations.
One unidentified senior diplomat was quoted as saying the new guidelines could “serve as a sort of mandate to do the deal”.
Mrs May is expected to heighten her warnings to Brussels of Britain’s readiness to walk away from the talks by holding a Cabinet meeting on Thursday to further discuss preparations for the UK quitting the EU without a deal.
Senior ministers are expected to spend three hours discussing contingency plans needed in case the Brussels talks collapse.
About 30 notes outlining the possible consequences of a no-deal Brexit for households and businesses are also set to be released on the same day.
Despite the encouraging tone from Brussels, opposition to the Prime Minister’s Brexit blueprint agreed at her Chequers country retreat last month appeared to be growing.
The number of Tories backing the Stand Up For Brexit campaign calling for her to “chuck Chequers” had risen to 25 yesterday after a flurry of new signatures over the weekend.
Ex-Brexit minister Steve Baker, a supporter of the campaign, warned the Tories could be torn apart if Mrs May presses ahead with the plan.
He said: “If we come out with her hoping to get Chequers through on the back of Labour votes, I think the EU negotiators would probably understand that if that were done, the Tory party would suffer the catastrophic split which thus far we have managed to avoid.”