Instead, the foreign secretary will focus on forging a UK-US free trade Brexit deal when he meets senior members of the president's team.
Britain is said to be ready to forget the spat after the US agreed not to repeat claims GCHQ wiretapped Mr Trump.
GCHQ said the claims made by the White House were "nonsense".
Mr Johnson will meet with congressional leaders and members of Mr Trump's team in Washington, including chief strategist Steve Bannon and senior adviser Kellyanne Conway.
Officials said he will not raise the diplomatic row which was sparked when White House press secretary Sean Spicer repeated allegations GCHQ spied on Mr Trump for Barack Obama.
The UK is said to be satisfied by reassurances from the US that the claims - described by GCHQ as being as "utterly ridiculous" - will not be repeated.
Officials hope Mr Johnson can carve out a role as a link-man between No 10 and the White House, says the BBC's diplomatic correspondent James Landale.
On his three-day visit to the US, the foreign secretary will attend a summit of countries fighting so-called Islamic State and chair a meeting of the UN security council.
He will also host a UN event on female empowerment around the world, aimed at giving women better access to schools, jobs and democracy.
Boris Johnson will avoid discussing claims the UK's intelligence agency spied on Donald Trump when he visits Washington later, officials say.