Foreign Secretary Dominic Raab has refused to close the door on the possibility of the trade talks with the European Union continuing past the Sunday deadline.
The Cabinet minister said "never say never" and highlighted that negotiations with Brussels have a tendency to "drag and drift".
Prime Minister Boris Johnson is expected to have a telephone call with European Commission president Ursula von der Leyen at midday after negotiations re-started in the Belgian capital at 8am UK time.
The two leaders had agreed during a dinner meeting last week that Sunday marked the deadline for a firm decision to be made on whether to continue the talks.
However Mr Raab, in comments made to broadcasters, suggested there could be room for manoeuvre on the end point of the negotiations over a free trade agreement.
He said there is "a long way to go" to find a resolution on the two outstanding issues of fisheries and so-called level playing field "ratchet" clause which would tie the UK to future EU standards.
But the former Brexit secretary added that discussions could continue past Sunday if Mr Johnson and Europe's top official are able to break the deadlock during their conversation.
The Cabinet minister told BBC One's Andrew Marr Show: "The bar is quite high for us to be able to keep talking.
"We would need at a political level a commitment to move on those two key issues.
"When you look at what I said at the start of the week, it is pretty similar to what I'm saying now, which is never say never because EU negotiations can often drag and drift.
"But actually we do need finality and therefore we need at the political level of Ursula von der Leyen, the president of the European Commission, that there is clarity the EU will move on those two key issues. If we get that then there are still talks to be processed."