Under the initial deal signed last week, Iran was to reopen the Strait of Hormuz, the key shipping channel through which 20% of the world’s oil and natural gas travels.
The US also agreed to lift a military blockade on ships going to and from Iranian ports.
The deal also includes a $300bn (£224bn) plan for Iran’s “reconstruction”, and the US terminating “all types of sanctions” on it.
But the issue of Iran’s nuclear programme is still to be negotiated.
On Sunday some vessels appeared to be entering, exiting and transiting the strait, according to location data on the maritime tracking website MarineTraffic, despite Iran’s claim – disputed by the US – to have closed the strait.
The initial deal also called for fighting to stop on all fronts, but in Lebanon Israeli air strikes have since killed at least 67 people, while Hezbollah attacks have killed five Israeli soldiers.
Israel has insisted that its conflict with Hezbollah is separate from the war on Iran, which it mounted alongside the US on 28 February.
Lebanon was drawn into the war shortly afterwards, when Iran-backed Hezbollah launched rockets into Israel in retaliation for a strike that killed Iran’s supreme leader.
Israel responded by launching a bombing campaign across Lebanon and occupying around 5% of the country’s territory in the south – hoping to drive back Hezbollah fighters from its northern border – and has said it has no intention of withdrawing.
Since 2 March, at least 4,106 people have been killed in Lebanon, the country’s health ministry says. Its figures do not differentiate between combatants and civilians.
Israeli authorities say 36 Israeli soldiers and four civilians have been killed on both sides of the border.
Additional reporting by Lana Lam, Emma Pengelly and Richard Irvine-Brown