Downing Street has said David Cameron stands by his criticism of Donald Trump but will work with whoever is elected US president.
The UK PM has called the Republican hopeful "stupid, divisive and wrong" over his call for a temporary ban on Muslims entering the US.
Mr Trump replied by warning he may not have a "very good relationship" with Mr Cameron if he enters the White House.
He is also involved in a spat with new London Mayor Sadiq Khan.
The US presidential contender said he would not forgive Mr Khan for calling him "ignorant" - and challenged the Mayor to take part in an IQ test, an offer mocked by Mr Khan's team.
Last year, Mr Trump, who has beaten his rivals to become the presumptive Republican candidate, called for a temporary halt to all Muslims entering the US in the wake of the deadly terror attack in San Bernardino, California.
He said many Muslims nursed a "hatred" towards America and a ban should be in force "until our country's representatives can figure out what is going on".
Responding to Mr Trump's comments at the time, Mr Cameron said: "I think his remarks are divisive, stupid and wrong and I think if he came to visit our country I think it'd unite us all against him."
Speaking on ITV's Good Morning Britain earlier on Monday, Mr Trump said: "It looks like we're not going to have a very good relationship, who knows.
"I hope to have a good relationship with him, but it sounds like he's not willing to address the problem either."