While officials choreograph the biggest divorce deal in EU history, it remains unclear whether May can get a deal through parliament, where opponents have said she is betraying Brexit by signing up the United Kingdom to EU subjugation.
"We're not quite there yet," Cabinet Office Minister David Lidington told BBC radio. "We are almost within touching distance now."
Asked if he was saying it was possible there could be a deal in the next 24 or 48 hours, he said: "Still possible but not at all definite, I think pretty much sums it up. Cautiously optimistic."
Sterling jumped half a percent to as high as $1.2917 on Lidington's comments.
The EU wants to get agreement on a draft deal by the end of Wednesday at the latest if there is to be a summit this month to approve it, although few in Brussels seemed to hold out much hope that a breakthrough could come this week.
Brussels diplomatic sources said the bloc was trying to cajole May into sealing a deal this week and rubber-stamping it later this month, fearing that any delay would increase the chances of rejection by her ministers or parliament.
Leaving approval until the next scheduled EU summit on Dec. 13-14 would mean British lawmakers would not vote on a deal until after Jan. 7 when they return from holiday.
"That's giving the opponents of the deal a lot of time to pull together a campaign to vote it down. That's why everyone on our side is pushing so hard for November," one source said.
The EU and the United Kingdom need an agreement to keep trade flowing between the world's biggest trading bloc and the fifth largest national economy.
But May has struggled to untangle nearly 46 years of membership without damaging trade or upsetting the lawmakers who will ultimately decide the fate of any deal she can secure.
'TEMPORARY AND NOT INDEFINITE'
With under five months until Britain leaves the EU, the so-called Northern Irish backstop is the main outstanding issue.
Asked if the UK could be trapped in a backstop against its will, Lidington said: "The prime minister has said again and again, if the backstop were ever to be used - we don't want it to be used - ... it's clearly got to be something that would be temporary and not indefinite."
But the intricacies of any deal, hammered out in late-night sessions at the European Commission's modernist Berlaymont building in Brussels, are unlikely to change the growing opposition to May.
By seeking to leave the EU while preserving the closest possible ties, May's compromise plan has upset Brexiteers, pro-Europeans, Scottish nationalists, the Northern Irish party that props up her government, and some of her own ministers.
"No one is fooled by this theatre. Delay after staged-managed delay," former foreign secretary Boris Johnson said. "A deal will be reached and it will mean surrender by the UK."
Johnson, a prominent Brexit campaigner who resigned from May's government in July over her strategy, said the deal would turn Britain into a colony of the EU.
Brexit Secretary Dominic Raab said he was confident of progress in talks, though he gave no further details.
Lidington, who voted to stay in the EU in the 2016 referendum, declined to say whether the proposed deal would make the United Kingdom wealthier or poorer, saying that the people had decided to leave the EU.
Asked whether Britain would have to start preparing in earnest for a "no-deal" Brexit if an agreement were not clinched by the end of Wednesday, as newspapers have reported, he said: "I'm not going to ascribe days to particular actions." (Additional reporting by Gabriela Baczynska; editing by David Stamp)
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