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.”
Mr Trump said he was not anti-Muslim but “anti-terror”, saying “we have a real problem and we have to discuss it”, as he called on Muslims to work with the people and “turn people in” that they suspected of extremism.
“I have many Muslim friends,” he said. “I was with one the other day, one of the most successful men, he’s Muslim and he said, ‘Donald you have done us such a favour, you have brought out a problem that nobody wants to talk about’.”
David Cameron’s official spokesman was asked about Mr Trump’s suggestion that they might not have a good relationship following the Republican candidate’s latest comments.
“The prime minister has made his views on Donald Trump’s comments very clear. He disagrees with them and I haven’t got anything further to add,” said the spokesman.
“He continues to believe that preventing Muslims from entering the US is divisive, stupid and wrong.”
The Number 10 spokesman said that Mr Cameron was “committed to maintaining the special relationship” whoever wins the presidential election.
“He has been clear that he will work with whoever is president of the United States,” said the spokesman.
No proposal had been made for a phone call between the prime minister and Mr Trump, but the spokesman said Downing Street would be willing to consider it.
Mr Trump also criticised what he called the “very rude statements” made about him by Sadiq Khan, after Mr Trump suggested he would make an “exception” to the ban for the London mayor.