Boris Johnson has called on European leaders to end the “doom and gloom” about Donald Trump’s election victory and see the US president-elect as someone with whom they can build closer ties.

Speaking after a phone conversation with the vice-president-elect, Mike Pence, the British foreign secretary described Trump as “a deal maker” and called for an end to the “collective whinge-o-rama” which followed Hillary Clinton’s defeat.

The British government is hurriedly seeking ways to engage with Trump, a man several ministers had condemned during his election campaign, including Johnson, who had said he was “genuinely worried” at the idea of a Trump presidency.

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One report on Friday claimed that the international trade secretary, Liam Fox, wanted to turn to Nigel Farage to liaise with Trump because the government had no links with his camp.

The story in the Telegraph was, however, immediately dismissed by Downing Street, with a spokesman saying: “Dr Fox has no plans to talk to Mr Farage.” A Ukip source said there was no truth to the report.

Facebook Twitter Pinterest Trump spoke with May on Thursday. Photograph: Andrew Kelly/Reuters

Theresa May talked to Trump on Thursday afternoon, with the president-elect making reference to the close relationship between Ronald Reagan and Margaret Thatcher, a Downing Street source said.

Trump “alluded to their relationship as a way to underline that he was keen to have a good personal working relationship too”, the source said.

Johnson said of his conversation with Pence: “We agreed on the importance of the special relationship and the need to tackle global challenges together.”

Speaking in Belgrade, where he had met the Serbian prime minister, Aleksandar Vučić, Johnson called for a sense of proportion in reaction to Trump’s success.

“I would respectfully say to my beloved European friends and colleagues that it’s time that we snapped out of the general doom and gloom about the result of this election and collective whinge-o-rama that seems to be going on in some places,” he said.

“He is, after all, a deal maker. He wants to do a free trade deal with the UK,” Johnson told reporters. “I believe that this is a great opportunity for us in the UK to build on that relationship with America that is of fundamental economic importance for us, but also of great importance for stability and prosperity in the world.”

Jonathan Marland, David Cameron’s former trade envoy, said Trump’s victory was a “great opportunity to rebuild alliances”.

Speaking on BBC Radio 4’s Today programme, the peer said: “Our relationship with America has been rocky in terms of trade recently because of the onslaught they had against our banks and against BP and I think business needs certainty. If Mr Trump can show there is a greater certainty and that he is open for business then I think it is very good for British business.”

He added: “Post-Brexit and post-Trump, both countries are going to be looking for allies with which to increase their trade. Both countries will be looking for quick-fix partners post these events and I have no doubt that Trump, whose mother was born in Scotland … will be looking very favourably on economic relationships with the UK.”

Facebook Twitter Pinterest Farage has been one of Trump’s biggest supporters in UK politics. Photograph: Ben Birchall/PA

Farage has spoken jokingly about the idea of being Trump’s liaison with the EU in Brussels, but his allies say that while he would happily consider being an intermediary for the UK government, there is no realistic chance he would be asked.

The Ukip interim leader is currently in Florida for a private event. In an interview with TalkRadio recorded before he was due to fly to the US, Farage joked about the idea of Trump sexually assaulting May. Imagining the two leaders meeting, he said: “Don’t touch her, for goodness sake,” before laughing. Asked about the likely behaviour of Trump, who has been accused of a series of sexual assaults, which he denies, Farage added: “If it comes to it, I could be there as the responsible adult role, to make sure everything’s OK.”