The United Arab Emirates and Bahrain signed in 2020, breaking a long-standing taboo to become the first Arab states to recognise Israel in a quarter of a century. Morocco and Sudan followed suit.
Mr Trump said: “In speaking to numerous of the great leaders mentioned above, they would be honoured, as soon as our document is signed, to have the Islamic Republic of Iran as part of the Abraham Accords. Wow, now that would be something special!”
Axios had earlier reported that his request was met with silence during the phone calls on Saturday.
The leaders of Saudi Arabia, Qatar and Pakistan do not have formal diplomatic relations with Israel, and Mr Trump’s suggestion was reported to have surprised them.
“There was silence on the line, and Trump joked and asked if they are still there,” a US official with knowledge of the call told the news site.
Mr Trump had earlier said he had urged his negotiators “not to rush into a deal, in that time is on our side”. He said any deal would “either be a great and meaningful one, or there will be no deal”.
Deal ‘has a lot of support’
Marco Rubio, the US secretary of state, told reporters during a trip to New Delhi: “We have what I think is a pretty solid thing on the table in terms of their ability to open up the [Strait of Hormuz], get the straits open.”
“It has a lot of support in the Gulf … every country that we’ve walked through it [with] understands it’s not just very reasonable, but it’s the right thing for the world to get done.”
Global oil prices fell following renewed optimism that Washington and Tehran would reach an agreement.
A US official said the US and Iran have agreed in principle to a deal that will reopen the Strait of Hormuz and involve Tehran surrendering its stockpile of highly enriched uranium.
On Monday night, the US launched “defensive” strikes in southern Iran, attacking what they said were missile launch sites and Iranian boats seeking to lay mines in the strait.
A US official insisted that the strikes did not indicate that the ceasefire was over.
Earlier that evening, Mr Trump posted to Truth Social that Iran must “either be immediately turned over to the United States to be brought home and destroyed or, preferably, in conjunction and coordination with the Islamic Republic of Iran, destroyed in place or, at another acceptable location, with the Atomic Energy Commission, or its equivalent, being witness to this process and even”.