Cabinet Office Minister David Lidington, effectively the UK's deputy prime minister, said it's "possible but not at all definite" that the two sides can reach an agreement this week.
"We are not quite there yet," Lidington told the BBC.
"This was always going to be an extremely difficult, extremely complex negotiation, but we are almost within touching distance now." May's Cabinet met Tuesday for an update on the talks, with the prime minister under pressure from pro-Brexit ministers not to make further concessions to the EU.
The prime minister said Monday evening that the negotiations were in their "endgame" but that she would not sign up to an "agreement at any cost." Britain wants to seal a deal this fall, so that Parliament has time to vote on it before the UK leaves the bloc on March 29.
The European Parliament also has to approve any agreement.ALSO READ: Brexit: How the Irish border became the biggest fight for EU and Britain
Negotiators have been meeting late into the night in Brussels in a bid to close the remaining gaps.
The main outstanding issue is how to ensure there are no customs posts or other checks along the border between the UK's Northern Ireland and EU member Ireland after Brexit.ALSO READ: Brexit: Four UK ministers on verge of quitting, EU rejects latest plan
The two sides are working on a proposed solution involving a common customs arrangement for the UK and the EU. But May faces pressure from pro-Brexit Cabinet members not to agree to an arrangement that binds Britain to EU trade rules indefinitely.
May also faces growing opposition from pro-EU lawmakers, who say her proposed Brexit deal is worse than the status quo and the British public should get a new vote on whether to leave or to stay.ALSO READ: Brexit pangs: JP Morgan leads $283 billion assets shift to Frankfurt