The foreign secretary has held “positive but frank” talks with some of Donald Trump’s key advisers during a visit to New York, officials have said.
Boris Johnson met Mr Trump’s son-in-law, Jared Kuchner, and Steve Bannon, the US president-elect’s strategist.
It was the first formal face-to-face meeting between members of the incoming US administration and a UK minister.
Officials said they discussed US foreign policy towards Syria, China and Russia.
On Monday Mr Johnson will be in the capital, Washington D.C, to meet key congressional leaders.
However, he will not meet his future opposite number Rex Tillerson.
Protocol says that cannot happen until the businessman is confirmed as secretary of state.
Mr Johnson’s visit comes hours after Mr Trump tweeted he is “very much” looking forward to meeting Theresa May in the spring.
He said that the prime minister, who has spoken to Mr Trump twice since the US election, would head to Washington to meet him after his inauguration.
He referred to Britain as a “long-time ally” and as “very special”.
By BBC diplomatic correspondent James Landale
Boris Johnson’s aim in New York is simple.
He is there to get Britain onto the front foot when it comes to its relations with the incoming administration.
From the moment Donald Trump was elected, the UK has been playing catch-up.
The president-elect caused a diplomatic storm when he decided that the first British politician he would meet would be Nigel Farage, whom he suggested would make a good UK ambassador to the United States.
The idea that the former UKIP leader was closer to Mr Trump than British ministers was hard for the Foreign Office and Downing Street to stomach.
So Theresa May sent her joint chiefs of staff over before Christmas.
Mr Johnson is now doing his bit, and then the prime minister herself will visit Mr Trump in Washington in the spring.
The reason this matters is because it is in the UK’s interests to stabilise UK-US relations so that Mrs May can urge Mr Trump against any rapprochement with President Putin of Russia while also seeking a post-Brexit trade deal.
Speaking on Sky News, Mrs May said Britain “shares values” with the US, but she denounced the president-elect’s previous obscene comments about women as “unacceptable”.
She said Mr Trump had himself apologised, adding: “But the relationship that the UK has with the United States is about something much bigger than just the relationship between the two individuals as president and prime minister.”
The foreign secretary will travel to Washington on Monday to meet senior republicans including House Speaker Paul Ryan, Senate Foreign Relations Committee Chairman Bob Corker and Mitch McConnell, the Senate majority leader.
Officials said the meetings will be to touch base with the incoming administration and establish a deeper relationship.
Mrs May received a phone call from Mr Trump two days after the US election and is thought to have been the ninth leader he spoke to.