Downing Street said there are still issues that need to be resolved as talks resumed today. But the Prime Minister and European Commission President Ursula von der Leyen are poised to talk to push an agreement over the line.
Irish premier Micheal Martin fuelled hopes of a breakthrough being made within days.
He said: “I would be hopeful that by the end of this week we could see the outline of a deal.
“That remains to be seen. It’s down to political will. One must remain hopeful that a deal can be arrived at.”
But Government sources warned the assessment was “optimistic”.
MPs and MEPs are facing emergency sittings to rush through any deal before the end of the year.
No 10 said Parliament has shown that it can “act at pace” when it needs to pass legislation swiftly.
Brussels is hoping to have a pact signed off by Monday to allow legislation to be passed in the European Parliament at a special sitting on December 28.
Talks resumed online today after they were brought to an abrupt halt last week when an EU official tested positive for coronavirus.
“We will take back control of access to our waters,” he added.
The spokesman insisted there will not be any extension to the transition period.
Brussels sources expect a last-minute trade-off on the so-called level-playing field and reduced access to Britain’s coastal waters to help get a deal across the line.
EU Parliament Brexit representative Christophe Hansen said the bloc would have to be prepared to meet Boris Johnson’s fisheries demands in order to clinch an agreement.
The Luxembourgish MEP said: “There will be compromises to be made on fisheries.
“The status quo, that is somewhere we’re not going to land.”
Bernd Lange, the EU Parliament’s trade chief, said: “It’s already five past midnight.
“We need a text, otherwise ratification and democratic scrutiny by the European Parliament will be a farce.”