Trump, in an interview with MSNBC, said Cameron extended the invitation to visit 10 Downing Street two days ago and that he "might do it."
He gave no other details, including when the visit might occur. Representatives of Cameron's office could not be immediately reached for comment to confirm the invitation.
London Mayor Sadiq Khan wants to 'educate' Trump on Islam
Trump's comments followed a transatlantic exchange between the two men over Cameron's comments criticising the New York real estate mogul's call for a temporary ban on Muslims entering the United States.
In December, Cameron called Trump "divisive, stupid and wrong" and suggested Trump would unite Britain against him if he visited the United Kingdom.
After Trump clinched his party's nomination this month, Cameron acknowledged the achievement but said he stood by his earlier comments and would not apologise.
Trump fired back this week, saying on Monday that he was likely not to have a good relationship with Cameron.
But the next day he said he expected to have "a good relationship" and on Friday Trump said Cameron had invited him to London.
"I will do just fine with David Cameron. I think he's a nice guy. I will do just fine," Trump told MSNBC on Friday. "But they have asked me to visit 10 Downing Street - and I might do it."
London's mayor an exception to proposed ban on Muslims: Trump
Trump defended his recent comments, saying he was not attacking Britain. As for Cameron, "he came after me a little bit," Trump added. Cameron has said he will work with the winner of the Nov 8 US presidential election and is committed to maintaining the special US-UK relationship, his spokesperson has said.
Trump's proposed ban also drew criticism from Sadiq Khan, who was elected mayor of London this month and is Muslim. Khan said last week that Trump's "ignorant view of Islam could make both our countries less safe."
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