Talks between the UK and the European Commission are taking place again this week, ahead of the EU27 leaders meeting to discuss Brexit at a summit in Salzburg, Austria, next week.
Barnier's comments were reported by the British embassy in Slovenia's Twitter account - which has just over 2,000 followers.
The pound jumped 0.73 per cent against the dollar following the tweet, leaving it up 0.9 per cent on the day at $1.3035.
It was also 0.38 per cent up against the euro to €1.1225.
Barnier has boosted the pound a handful of times in recent weeks, after traders took heart from apparently renewed optimism from the Commission's man in charge of Brexit talks.
The pound rose last week and the week before after he talked up an "unprecedented" deal.
Craig Erlam, senior market analyst at Oanda, said: "Sterling has become very sensitive to positive Brexit news over the last couple of weeks, having spiked on a couple of occasions on reports that the UK will get a bespoke deal and that Angela Merkel is willing to accept less detail on future ties. Clearly there is a feeling that a lot of Brexit pessimism and no deal risk has been priced in which is why we’re in a state of such sensitivity to any reports that indicate a breakthrough will come.
"The reported comments do certainly support the previous reports and provide some hope that both sides are fully committed avoiding a no deal scenario, something that at times hasn’t always appeared the case. Assuming these comments aren’t denied and this is the case, it’s possible that things could be really looking up for the pound after what has been quite an awful summer for the currency, having fallen more than 10 per cent against the dollar at one stage.
"The pound is also vulnerable to these comments being clarified or a caveat being attached that a no deal is also still very possible, as we’ve seen when previously – apparently – positive reports appeared."
Connor Campbell, financial analyst at SpreadEx, added: "Once again a Brexit-inspired movement from the pound came to dominate an otherwise quiet afternoon session.
"If one person can shift sterling at the moment it is Michel Barnier. The currency is desperate for any signs of good news from the EU’s chief negotiator, often making the most of some pretty tepid statements. Monday’s gains seem slightly more justified however, with Barnier saying that it was a ‘realistic’ possibility a deal could be reached by November. And though, of course, the content of any deal is the thing that really matters, at the moment the pound will take what it can get."
Chris Leslie MP, a leading supporter of the People’s Vote campaign, said: “The current volatility of sterling just demonstrates how the markets are crying out for some certainty in the Brexit process.
“The Prime Minister knows she can’t get her botched Brexit proposals past Parliament, so instead she’s trying to force through a ‘blindfolded Brexit’, where we leave the EU with no idea of where we’re going to end up. That’s not good enough. That’s not the Brexit that anyone voted for in 2016.
“Despite the vagaries of the market, the negotiations remain stalled and the risk of a chaotic Brexit is increasing. That’s why more and more people across the country are calling for a People’s Vote.”