Mediators have established communications channels to de-escalate any incidents, and technical talks are set to continue, the source, with knowledge of the discussions, added.
A senior Iranian source said there would be a meeting in Doha on Tuesday, but unlike previous technical talks between Tehran and Washington in Switzerland, the focus would be on managing the Strait of Hormuz and de-escalating tensions.
US President Donald Trump confirmed in a social media post that there would be a meeting in Doha on Tuesday, but gave no details.
The US and Iran signed a 14-point memorandum of understanding aimed at ending four months of conflict on June 17, under which both sides agreed to cease hostilities and reopen the strait, through which a fifth of the world's oil and liquefied natural gas typically transits.
Closure of the waterway sent oil prices to above $US100 ($A145) a barrel, pushing up global inflation and causing a political headache ahead of midterm elections for Trump, who touted falling oil and gas prices in a series of other early morning posts on Monday.
The accord paves the way for 60 days of more in-depth talks on thornier issues such as Iran's nuclear program, although both sides have given conflicting accounts as to what was agreed.
Iranian President Masoud Pezeshkian said on Monday that half of $US12 billion ($A17 billion) of assets frozen in Qatar would be released following the accord and returned to Iran, Iranian state media reported.
He described the memorandum, which includes waivers for sanctions on Iran's oil and petrochemical sectors, as "a great victory for the Iranian people."
The senior Iranian source said Doha and Tehran were in the final stages of agreeing on technicalities for the release of the first $US6 billion ($A8.7 billion) of frozen assets, which he said would be issued in two tranches.
Oil prices rose on Monday with Brent crude futures up almost one per cent after the weekend tension highlighted the fragility of the US-Iran accord.
White House Press Secretary Karoline Leavitt told Fox News on Monday that US envoys Steve Witkoff and Jared Kushner would fly to Doha this week for high-level meetings, which would take place alongside technical talks.
"As far as we're concerned, we're holding up our end of the ceasefire. Violence will be met with violence," Leavitt added.
But in an indication of the uncertainty around the interim deal, Iran's Deputy Foreign Minister Kazem Gharibabadi said on Monday, before Trump's post, that technical working group meetings were not scheduled for this week, according to Iran's Tasnim news agency.
French President Emmanuel Macron on Monday said he was working with Oman to de-escalate tensions in the Middle East and would cooperate with partners to de-mine the Strait of Hormuz.
"We have decided to collaborate, in conjunction with our partners, on clearing mines from the Strait in order to secure maritime routes and guarantee free and unconditional passage through the Strait of Hormuz," Macron wrote on X after meeting with Oman's Sultan Haitham bin Tariq al-Said at the Elysee Palace.
A return to talks would follow several days of strikes and counterstrikes since an Iranian projectile hit a cargo vessel in the Strait of Hormuz on Thursday, with both the US and Iran accusing the other of breaking the interim ceasefire.
Iran, which has sought to exert its authority over shipping in the strait, launched missiles and drones at US military sites in Kuwait and Bahrain early on Sunday.
Iran's Islamic Revolutionary Guard Corps said US strikes had violated the ceasefire and that American bases in the region would "experience hell in the coming days".
