The European Union suggested there is a "narrow path" to a deal with the UK, but that the next few days will be critical.
The positive soundings from the EU helped to nudge the pound above $1.35 against the dollar and raised hopes both sides could overcome their differences.
EU chief Ursula von der Leyen said she could not say whether there would be a deal, but insisted there was a path to get there.
She said: "The path may be very narrow but it is there. We have found a way forward on most issues but two issues still remain outstanding: the level playing field and fisheries.
"I am glad to report that issues linked to governance now have largely been resolved. The next days are going to be decisive.”
Britain and the EU are in the final stretch of tortuous talks to safeguard an estimated $1 trillion of free trade beyond December 31, when the UK completes its transition out of the world's largest trading bloc.
Ms Von der Leyen said discussions about access to UK fishing waters for EU vessels were "still very difficult" but negotiators had moved forward on the other most contentious element – guarantees of fair competition for companies.
"On standards, we have agreed a strong mechanism of non-regression. That's a big step forward. And this is to ensure that our common high labour, social and environmental standards will not be undercut," she said.
"Difficulties still remain on the question of how to really future-proof fair competition. But I'm also glad to report that issues linked to governance by now are largely being resolved."
She said she sometimes felt Britain and the EU would not be able to overcome their differences, but they would keep trying.
Failure to get a deal would damage economies on both sides of the English Channel, send shock waves through markets, snarl borders and cause chaos in supply chains across Europe as it struggles with Covid-19.
But the upbeat mood around a possible Brexit deal before the end of the year has helped the pound strengthen more than 2 per cent against the dollar this week.
But caution still shows in elevated implied volatility gauges, which imply that traders are buying protection against price swings in the currency.
Tempering some of the optimism were comments from a British official, who said both sides had made some progress in the trade talks, but were still "very far apart in key areas".
At opening trading, sterling was up 0.3 per cent against the dollar at $1.3507. It rose above the $1.35 mark shortly after Ms von der Leyen's comments.
Against a broadly stronger euro, however, it was down 0.2 per cent, making a euro worth 90.34 pence.