The UK and its European allies have until 12 May to persuade the US president to stick with the deal, BBC reported.
Trump has criticized the agreement, which he calls "insane".
In a call with Theresa May on Saturday, the president "underscored his commitment to ensure that Iran never acquires a nuclear weapon".
In the landmark deal – signed by the US, China, Russia, Germany, France, the UK and Iran – the latter agrees to limit its nuclear activities in return for the easing of sanctions on its economy.
UN also warned Trump not to walk away from the deal.
But Trump has threatened to withdraw unless the signatories agree to "fix the deal's disastrous flaws".
Johnson will meet US Vice-President Mike Pence, National Security Adviser John Bolton and foreign policy leaders in Congress.
Ahead of the trip, Johnson said the UK and US are "in lockstep" on many global foreign policy issues, citing the response to the use of chemical weapons in Syria and the Salisbury poisonings.
Trump has until the deadline of 12 May to make a decision on the deal - the next deadline for waiving sanctions.
Johnson's discussions are also expected to cover the crisis in Syria and also North Korea, ahead of Trump's meeting with Kim Jong-un, which now has a date and location arranged.