A No 10 source describes the negotiations as “positive”.
King Charles had also been due to meet the president of the European Commission in the UK on Saturday, the BBC understands.
Multiple sources said the visit was cancelled due to operational reasons unconnected to the political talks.
The planned meeting between the King and Ursula von der Leyen, originally reported by Sky News, was not part of the negotiations between the UK and the EU.
But the fact a meeting was planned is significant as it appears to indicate a Brexit deal was about to be done – and publicly presented – while Ms von der Leyen was in the UK.
It is not known when the European Commission president will now come to the UK.
Some had suggested a new deal on the Northern Ireland Protocol – thought to be all but complete – could be called the Windsor Agreement and include a moment in front of the cameras involving Prime Minister Rishi Sunak and Ms von der Leyen.
The protocol, which was agreed under former Prime Minister Boris Johnson and came into force in 2021, saw Northern Ireland continue to follow some EU laws to get round the need for checks at the UK’s border with the Republic of Ireland.
Mr Sunak and Ms von der Leyen spoke by telephone on Friday and Downing Street said the prime minister had made “good progress”.
A source said afterwards that it had been “positive” and negotiations would continue, with the leaders agreeing “to discuss this further in coming days.”
Both the UK and the EU have to coordinate diaries to make the choreography of an announcement work at a mutually convenient time and place.
Irish Taoiseach Leo Varadkar told reporters on Saturday that a deal was not done yet on the Protocol but was “inching towards a conclusion”.
“There is the possibility of agreement in the next few days but by no means guaranteed… there’s still a gap to be closed,” the premier said, according to Reuters news agency.
Meanwhile, Foreign Secretary James Cleverly has pulled out of a long-planned trip to the Middle East at the beginning of next week – raising expectations that a formal announcement from the UK and Brussels could be days away.
But there have been repeated delays over the past week or so, as wrangling went on between No 10, the Democratic Unionist Party and Conservative backbenchers.
The prime minister has been trying to win support for changes to the controversial protocol.
Earlier, a source from the DUP told the BBC they had not been involved in any talks with the prime minister on Friday and had no meetings scheduled over the weekend.
Mr Sunak has been trying to win over the DUP to a deal, as the party is currently blocking the formation of devolved government in Northern Ireland.
Sammy Wilson MP, the DUP’s shadow spokesperson for Brexit, said his party would only accept a deal if EU law “imposed” on Northern Ireland was removed.
He told Sky News: “If the prime minster succeeds in getting that we will embrace it, but if he hasn’t succeeded in achieving that aim then as part of the UK we cannot accept it.”
Mr Wilson said that “we expect British law should apply in Northern Ireland, not Brussels law imposed on us, with all the disastrous effects that has.”
The prime minister has been facing pressure from some Conservative MPs over Northern Ireland’s current obligation to follow some EU laws and be accountable to the European Court of Justice.
Eurosceptic Tory MP Sir John Redwood said: “The UK needs to hold out over the EU imposing laws on Northern Ireland. The EU needs to get the Unionists on side.”