0612 GMT June 24, 2019
The pair, in a phone conversation, also spoke of the need to fight the spread of terrorism and establish safe zones in war-ravaged Syria and Yemen, the White House statement read.
No further details were provided about those plans, AFP reported.
The official Saudi Press Agency early on Monday confirmed that Trump had called Salman.
It made no mention of Iran but said the views of the two leaders "were identical" on issues discussed during the call.
SPA said Trump and Salman also agreed on "formulating the appropriate mechanisms" for countering "terrorism" and extremism.
Trump opposed the nuclear agreement signed by Iran and world powers, including the United States, in 2015 and has said he wants to undo it.
Some of his key nominees have adopted an openly anti-Iran stance, including secretary of state candidate Rex Tillerson, who is seeking a complete revision of the accord.
But before he left office, former president Barack Obama warned against rowing back the pact, emphasizing its "significant and concrete results".
The deal put limits on Tehran's nuclear program in exchange for the lifting of international sanctions.
Salman and Trump invited each other to visit their respective capitals, the Saudi Press Agency said.
"The two leaders agreed to schedule the visits in the coming period", it said.
The United States and Saudi Arabia have a decades-old relationship based on the exchange of American security for Saudi oil.
But ties between Riyadh and Washington became increasingly frayed during the eight-year administration of Obama.
Saudi leaders felt Obama was reluctant to get involved in the conflict in Syria and other regional conflicts.
Saudi Arabia, where the extremist Wahhabi ideology is practiced, is widely believed to support and fund the Takfiri terrorist groups that have been wreaking havoc in Iraq and Syria.
Riyadh's Foreign Minister Adel al-Jubeir has said he expects the Trump administration to be more engaged in the Middle East, and the world in general, while "rebuilding" relationships with allies.