Mr Johnson has jetted to NY to meet with senior members of Mr Trump's team in a bid to pave the way for a face to face meeting between the president and Prime Minister Theresa May in the spring.
But on Sunday in an interview with Sky News Mrs May made it clear that she thought Mr Trump's comments about groping women were "unacceptable".
Johnson met Trump's son-in-law, Jared Kushner, and his chief strategist, Steve Bannon, for "positive and frank talks" in NY.
On Monday, the Foreign Secretary, one of the main cheerleaders for Brexit, will head to Washington to meet influential Republicans, which is likely to include Congressional leaders such as Speaker Paul Ryan, Senator Bob Corker and Senator Mitch McConnell.
Johnson, a leading member of the campaign for Britain to leave the European Union, is likely to discuss the status of the "special relationship" between the two countries once Trump takes office and after the United Kingdom leaves the bloc.
The Government has been seeking to forge links with the incoming Trump administration, and Mr Johnson's visit follows meetings between the president-elect's team and Mrs May's closest advisers Nick Timothy and Fiona Hill.
The Foreign Secretary is understood to have met Jared Kushner, Trump's son-in-law and senior adviser, and Steve Bannon, his chief strategist, in NY.
Johnson has been less than complimentary about Trump in the past, once remarking that "the only reason I wouldn't visit some parts of NY is the real risk of meeting Donald Trump".
The NBA Unveiled Their Perfectly Fine 2017 All-Star Game Jerseys
Milwaukee's Giannis Antetokounmpo now has 500,663 votes and looks poised to make his first All-Star appearance this season. Warriors center Zaza Pachulia leads Kawhi Leonard and Anthony Davis among frontcourt players in the West All-Star voting.
The President-elect took to Twitter this morning to say he was looking forward to meeting May, and described Britain as "very special".
It was announced on Friday that Prime Minister Theresa May, who has twice spoken to the president-elect since his victory over Hillary Clinton last November, is to travel to Washington DC to meet Trump next month.
Farage is also attending Trump's inauguration. "The conversations I've had, I think we're going to look to build on that relationship for the benefit of both the United States and the U.K. and I think that's something that's optimistic and positive for the U.K. for the future".
"I think that is something that is optimistic and positive for the United Kingdom for the future".
Reince Priebus, the former Republican National Committee chairman, said Mr Trump understands that Moscow was behind the intrusions into the Democratic Party organisations.
Johnson had been extremely critical of Trump before he became the president-elect. Although Johnson has knocked Trump, he has also defended him from criticism in Europe.
She added: "Sometimes people think that there is going to be a change in the American approach to North Atlantic Treaty Organisation".