But it’s not clear Trump’s pressure will win him the nationalist ally he craves in Europe.

For weeks, the discussion about May’s replacement has centred on Boris Johnson, the voluble Brexit leader and former foreign secretary who shares Trump’s penchant for headline-grabbing quotes.

On Tuesday, however, it was Labour Party leader Jeremy Corbyn who grabbed the spotlight.

Corbyn had turned down a visit to the lavish state banquet at Buckingham Palace the night before saying that the US president did not deserve the honour.

Instead, he delivered an anti-Trump screed at a massive rally less than mile away from the president’s joint press conference with May.

And, to cheers from the crowd, he attacked Trump over his policies on immigration, climate change, reproductive rights and international cooperation.

Asked about the comments at the press conference, Trump said Corbyn had asked him to meet, “and I told him no”.

“I don't know Jeremy Corbyn. Never met him, never spoke to him,” Trump added.

The back-and-forth had both sides suddenly contemplating the potential US-UK relationship under a Prime Minister Corbyn – a distinct possibility if a general election proves the only way to resolve the UK's political impasse over Brexit.

The US and the UK have long proclaimed a “special relationship” exists between the two countries. Trump used the phrase liberally and warmly during his remarks, proclaiming “the greatest alliance the world has ever known”.

Corbyn’s rise to power would almost certainly further test the relationship’s strength still further.

Senior Labour officials war-gaming the prospect of Corbyn as prime minister with Trump still in the White House anticipate a rocky relationship, but not a total breakdown.

A diplomatic visit like this week would still take place, one official said, but not with full state honours. Corbyn opposed the granting of a state visit and would continue to do so as prime minister.

“If he were president and Jeremy were prime minister they would find a way of working together. While of course simultaneously being a vocal critic,” the official said.

Whether Corbyn would call the US relationship “special” – a buzzword for post-war UK prime ministers – remains to be seen.

He used the term in a 2017 conference speech, but did so with conditions. He warned that Trump had “threatened war and talked of tearing up international agreements”.

“If the special relationship means anything, it must mean that we can say to Washington: that way is the wrong way,” Corbyn added at the time.

Some of Labour’s criticisms of Trump – on the Iran nuclear deal, on climate change, on the importance of multilateral bodies like the United Nations – are shared by the Conservative government, and are making the transatlantic relationship more difficult.

Heather Conley, director of the Europe programme at the Centre for Strategic and International Studies predicted the US-UK relationship would face “profound strain” if Corbyn were to occupy No. 10 Downing.

For one, the policies he has espoused over his lengthy political career are “so dramatically different from what we've come to expect from the United Kingdom,” she said.

“The more I think through what a Jeremy Corbyn premiership would look like, I think it would almost have some similar aspects to what it looked like when Trump became president,” Conley said.

“You would really have a challenge for civil servants who wish to prevent dramatic shifts in policy.”

Corbyn as prime minister would add major new areas of divergence, particularly on military cooperation.

Corbyn opposed last year’s air strikes in Syria, as well as the 2011 Libya intervention.

He can be counted on to maintain his lifelong opposition to US foreign interventions if he became prime minister.

It is also impossible to imagine Corbyn would be an ally, as May has been, in Trump’s push for Nato allies to spend more on defence.

“I imagine everybody would notice we have a tangible change. We have different policies and different priorities from the current government,” the Labour official said.

Then again, the official added: “We could have Bernie”.

The left-wing Democrat presidential hopeful would likely be a much more comfortable fit for a Corbyn premiership.

Trump allies have trouble envisioning Corbyn as prime minister, mainly because they don’t believe it will happen.

Asked whether Trump would meet Corbyn at the negotiating table if he becomes prime minister, the White House declined to comment.

Trump also did not provide a reason for rejecting Corbyn’s meeting request.

The president spent his time Tuesday engaging conservative politicians – one of whom, following the party’s leadership election, will become prime minister.

Johnson, one of the leaders of the official Vote Leave campaign during the Brexit referendum, further bolstered his hardline Brexit credentials when he resigned from May’s Cabinet in protest at her concessions to the EU.

The president's team also requested a sit-down meeting with Environment Secretary Michael Gove, another leader of the official Vote Leave campaign during the Brexit referendum.

But with Trump and Corbyn’s public spat on Tuesday, the president’s allies were forced to confront the possibility of a future Corbyn-led Britain.

One ex-Trump aide suggested the Labour leader isn’t particularly popular within his own party after it suffered heavy losses in last month’s European Parliament election.

“He’s as well-liked within his party as Hillary Clinton was with Democrats” in 2016, the ex-aide said.

The White House has been careful not to press too hard on issues where May’s successor – whether it’s Johnson, or a handful of other potential candidates – might diverge from Trump, like health care and climate change.

Trump was noticeably quiet on the latter issue during his press conference with May.

He also didn’t make waves after discussing the topic over tea on Monday with Prince Charles, a major proponent of climate rules and the Paris climate accord.

On trade though, Trump was typically robust.

“When you are dealing with trade everything is on the table, so NHS [National Health Service] or anything else … but everything will be on the table, absolutely,” he said.

May quickly pointed out that what was included in a scope of any deal would be the subject of negotiation.

But if the UK is still in a state of political upheaval and desperate to make up for trade losses with the EU, those talks may prove to be a painfully one-sided affair – special relationship or not.