The Elysée Palace said on Wednesday it did not comment on reports and added that there were “no planned meetings” between Mr Macron and Mr Johnson in the near future.

Earlier in the day, the Prime Minister’s official spokesman stressed the importance of collaboration, saying: “The important thing I think the public on both sides of the Channel want us to do is to focus on how we avoid further loss of life.”

Asked about Sylvie Bermann, a former French ambassador to Britain, who this week declared Anglo-French relations at their lowest ebb since the Battle of Waterloo in 1815, the No 10 spokesman said: “We continue to have very close relationships with the French government and our French counterparts. And we will continue to work with them on a number of issues including on tackling the issue of small boats.”

It is understood that a letter expected from French prime minister Jean Castex setting out French proposals for a new post-Brexit deal on migration with the EU had not arrived by Wednesday night.

Gérald Darmanin, the interior minister, said earlier this week that France was ready to begin discussions if the British entered talks in a “serious spirit”.

The French are expected to offer a deal that would see Britain allowed to return one migrant for every child refugee they agree to take in from Calais. That is likely to be rejected by the UK as not going far enough in deterring migrants from attempting the dangerous 21-mile journey across the Channel.

The UK wants France to agree to take back all migrants who cross the Channel, which would destroy the people smugglers’ business model. In return, the UK would offer to take some unaccompanied children with family ties to Britain.

Downing Street insisted that a returns agreement, as set out by Johnson in his letter on Friday, would be the “single biggest deterrent” to migrants trying to cross the Channel.

Priti Patel talks to Italy in bid to outflank the French

On Thursday, Priti Patel will meet her Italian counterpart in Rome as part of a series of bilateral one-to-ones with EU ministers that will be seen as an attempt to outflank the French.

The Home Secretary will meet Italian interior minister Luciana Lamorgese to share tactics as both countries have used similar approaches to combat the surge in migrants.

Italy has seen migrant crossings from north Africa double to 63,000 this year while the UK has seen them triple to more than 25,000.

Italy and the EU have handed the Libyans millions to intercept migrants, used anti-Mafia laws to crackdown on traffickers and informally had “turn back” and “return” deals with Libyans.

Ms Patel spoke with her Dutch counterpart on Sunday, met with the Albanians on Wednesday and has meetings pencilled in with the EU Commission and Germans.

The moves represent an attempt to elevate the issue after efforts to find a bilateral solution to the Channel migrant crisis with France failed.

Ms Patel is canvassing for toughening border controls and seeking a wider agreement with the EU to take back Channel migrants who reach the UK after passing through “safe” European countries where they should have claimed asylum.

She hit out against the “appalling trade in human cargo or the weaponization of people”, adding: “Countries across Europe must come together to respond and strengthen both our resolve and borders to disrupt this trade, and that is exactly why I am in Italy today.”

Mr Macron on Wednesday risked deepening the cross-Channel row by urging Brussels “not to cave” in tense negotiations over the Northern Ireland Protocol.

He called for EU negotiators to maintain customs checks on goods shipped to the province from Great Britain in order to protect the bloc’s Single Market, and suggested they should stand firm against their British counterparts, like he has in the long-running spat over fishing rights.

“This protocol is of existential importance for Europe not to cave in when it comes to the integrity of our Single Market, otherwise there will no longer be any rules at our borders ” Mr Macron told EU regional leaders.

“It’s a question of war and peace for Ireland, so we should avoid any temptation to be less than serious.”

Mr Macron said the EU should maintain “solidarity”. “Whether it’s fisheries, the Irish agreement or migration,” he added.

“These are all issues between the EU and the United Kingdom.”

Lord Frost, the Brexit Minister, will now hold virtual talks with EU counterpart Maros Sefcovic, after the pair shelved plans for a face-to-face meeting in Brussels on Friday because of the omicron variant.

An EU source confirmed the pair would continue their discussions over the Northern Ireland Protocol from afar.