UK Prime Minister Theresa May is to visit President-elect Donald Trump in the USA next month after he takes office on 20 January 2017.
Confirming the Washington visit by Ms Hill and Mr Timothy, a Number 10 source said it was part of the build-up to the Prime Minister's first trip to meet the new President. "President-elect Trump agreed this would be useful".
Mr. Timothy tweeted that he didn't want anyone "reaching out" to Mr. Trump, while Ms. Hill tweeted: "Donald Trump is a chump".
During the call, May's second such conversation with Trump since his election on November 8, the prime minister "suggested it would be a good idea for key staff from both teams to meet".
He said: "It is entirely appropriate for the British Prime Minister and the American President, whoever they are, whatever parties they are, to have a good working relationship".
In the days after his election, Mr. Trump suggested that Mr. Farage might be best placed to act as a go-between for the two administrations, an idea which Downing Street firmly rejected.
May at the time emphasised her wish to strengthen trade and investment with the U.S. as the United Kingdom leaves the European Union following a June 2016 referendum on membership of the bloc.
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The prime minister's office said the visit was secured after her aides met with Trump's team.
Trump has struck a more positive note, although he has expressed scepticism about trade deals more generally, pledging to withdraw from the Trans Pacific Partnership with Japan, for example, and to throw up new barriers against imports.
The groundwork for the meeting was laid last month when Mrs May sent her joint chiefs of staff on a secret mission to NY and Washington.
Mrs May is keen to build relations with Mr Trump after describing his plan to ban all Muslims from entering the U.S. as "plain wrong" when she was Home Secretary.
May's predecessor, David Cameron, met President Barack Obama in March 2012 following his second inauguration that January.
The first phone call between the two was made following Trump's election.
"You would expect a meeting between the new US president and our prime minister relatively soon after he takes office and that is what is happening", he said. On that call, he invited the premier to visit Washington as soon as possible.