After weeks of intense London-Brussels talks, momentum has been building towards an agreement to revise the Northern Ireland Protocol - the arrangements agreed to avoid a hard border with EU member Ireland when Britain exited the EU in 2020.
Conservative lawmakers have been warned by party officials that there is a "three-line whip" to be in parliament on Monday, their strictest order for politicians to be present or they could face disciplinary action.
Cabinet ministers have been put on alert for a possible conference call over the weekend to discuss the deal, according to The Times newspaper, though that would depend on the backing of the biggest unionist party in the British province.
A government official, who asked not to be named, stressed that talks were continuing and said that any discussion of a conference call on Sunday was "purely speculation".
Sunak had a "positive" call with EU Commission President Ursula von der Leyen on Friday, a source at his office said, adding there had been "good progress" and the pair would discuss the issue further in coming days.
The prime minister also met with British retail executives on Friday to discuss the protocol, where the government's approach was reasonably well-received, according to a person with knowledge of the meeting.
The sense was that the government was trying, but no one was going to get everything they wanted, they added.
Meanwhile Sky News reported that Downing Street had planned a meeting on Saturday between Britain's King Charles and von der Leyen, which the broadcaster said could have suggested he was giving his blessing to the negotiations, or to endorse a deal had one already been agreed.
However, the meeting was cancelled on Friday, Sky said. Such a meeting would have been controversial as constitutionally the British monarch is not supposed to be involved in political issues.
Neither Downing Street nor Buckingham Palace responded to requests for comment about the Sky report.
Britain and the EU have been working to revise the protocol which would mark an end to a two-year standoff, but might lead to a new battle for the government with pro-Brexit Conservative lawmakers.
The government needs the support of the Democratic Unionist Party (DUP) if a deal is to restore Northern Ireland's power-sharing devolved administration, which its lawmakers are boycotting over their opposition to the protocol.
Some of Sunak's lawmakers will refuse to support the deal unless it has the backing of the DUP, which argues the imposition of checks on some goods coming from the rest of the United Kingdom is undermining the union with Britain.
The party has proven to be a central player in almost seven years of often tortuous Brexit talks and its resistance has torpedoed previous attempts at agreement.
Britain's foreign minister James Cleverly said on Friday he hoped any deal, if it were sealed, would address all the conditions laid down by the DUP.
"When, hopefully, we get those issues resolved then I would hope that the DUP would recognise that we've addressed their concerns and until we have addressed those concerns we're not going to sign off on the deal," he said.
Naomi Long, the head of the province's cross-community Alliance Party, said it was wrong to give any individual party what appeared to be a veto on Northern Ireland's future.
A DUP spokesperson said the focus in London and Brussels should be on getting the right deal, not rushing ahead.
"The wrong deal will not restore power-sharing but will cement division for future generations," the spokesperson said.
Opinion polls have consistently shown a majority of Northern Irish voters - who earlier opposed Brexit - favour the idea of the protocol and it is also backed by a majority of lawmakers elected to the devolved assembly last year.
The latest quarterly poll conducted for Queen's University Belfast showed on Friday that 53% see the current lighter touch application of the rules as an appropriate means of managing Brexit, down one percentage point in the last three months.