A "narrow path" has opened up for the UK and EU to strike a post-Brexit trade deal, the president of the European Commission has said.
Ursula von der Leyen said the "next few days are going to be decisive", with just two weeks left before the UK quits EU trading rules.
She said differences over enforcing a deal are "largely being resolved," but talks over fishing remain "difficult".
PM Boris Johnson told MPs there was "every opportunity" to reach a deal.
Officials from both sides are continuing talks in Brussels, as they race to strike a deal before the UK's post-Brexit transition period ends on 31 December.
Despite weeks of intensive talks, they have remained stuck over fishing rights and how far the UK should be able to depart from EU rules.
Updating the European Parliament on an EU leaders' summit last week, Mrs von der Leyen said: "As things stand, I cannot tell you whether there will be a deal or not.
"But there is a path to an agreement now - the path may be very narrow, but it is there."
She said that negotiators had agreed a "strong mechanism" to ensure neither side lowers their environmental or social standards, which was a "big step forwards".
But she added differences remained over how to "future proof" rules in this area, although disagreements over how to enforce a deal "by now are largely being resolved".
On Wednesday Mr Johnson's spokesman said: "We have made some progress in some areas, but it still remains that there are some significant gaps."
He added that it is "still the case" the prime minister views no deal as "the most likely outcome".
- Brexit happened but rules didn't change at once: The UK left the European Union on 31 January 2020, but leaders needed time to negotiate a deal for life afterwards - they got 11 months.
- Talks are happening: The UK and the EU have until 31 December 2020 to agree a trade deal as well as other things, such as fishing rights.
- If there is no deal: Border checks and taxes will be introduced for goods travelling between the UK and the EU. But deal or no deal, we will still see changes.
Speaking after Mrs von der Leyen, Mr Johnson said: "There's every opportunity, every hope I have, that our friends and partners across the Channel will see sense and do a deal.
"All that it takes is for them to understand that the UK has a natural right, like every other country, to be able to want to control its own laws and its own fishing grounds."
He told MPs at Prime Minister's Questions that "whatever happens in the next few days," the UK will "prosper mightily" whether a deal is found or not.
Progress on subsidies
Mrs von der Leyen also reported progress in another area which has proved contentious - agreed rules on how and when each side can give government subsidies to private firms.
She confirmed the two sides were now trying to agree "common principles" for when subsidies could be offered.
At an earlier stage in talks, the EU had insisted the UK should follow its current and future "state aid" rules in this area - a demand rejected by the UK.
The German politician added that there had been progress on "guarantees of domestic enforcement" of the rules, as well as allowing both sides to "autonomously" take action where disagreements arise.
Fishing troubles continue
However, she was more downbeat on fishing, where the two sides are haggling over access to each other's waters for their fishermen after 1 January.
"In all honesty, it sometimes feels that we will not be able to resolve this question," she said, but added that continuing the talks was the "only responsible" course of action.
She added the EU respected British "sovereignty" over its waters, but needed "predictability and stability" for European fishing fleets.
Meanwhile, it has been announced that both Houses of Parliament will begin their Christmas recess at the end of Thursday's sitting.
But No 10 said MPs and peers could be recalled to Westminster to vote on legislation to implement a deal before the end of the Brexit transition.
On Tuesday, Commons leader Jacob-Rees Mogg said Parliament would ideally need six days to pass any such law, but this period could be "truncated" if required.
Any potential deal would also need to be voted on by the European Parliament and potentially EU national parliaments before it can fully come into force.
EU leaders can in theory decide to provisionally apply any agreement and hold these votes after 31 December, but it would be unpopular among MEPs.