Iran has said it has replied to a US peace proposal, on a day when a month-old ceasefire showed signs of fraying, with drone strikes reported around the region.

Iranian state media reported that the Iranian response had been passed to Pakistani mediators, without giving further details. The announcement came a week after the US presented a peace proposal, which was reported to consist of a one-page 14-point memorandum of understanding that would reopen the strait of Hormuz while setting a framework for further talks on Iran’s nuclear programme.

The memorandum was itself a response to an earlier Iranian proposal. That also envisaged the lifting of parallel US and Iranian blockades on the strait, which have been driving up oil prices and stifling the global economy, with an emphasis on the lifting of sanctions and the release of frozen Iranian assets.

The announcement of a new response from Tehran came on a day when the Pakistan-brokered ceasefire, which came into effect on 8 April, showed new signs of strain. The United Arab Emirates and Kuwait reported drone incursions in their airspace on Sunday, and a drone attack started a small fire on a ship on the coast of Qatar.

Another drone strike was reported at a camp used by an Iranian Kurdish rebel group near Erbil in north-eastern Iraq.

Qatar’s defence ministry did not give details of the vessel targeted on Sunday, other than that it had come from Abu Dhabi. The UAE defence ministry said it had shot down the drones which entered its airspace, and which it said were Iranian.

Kuwait’s defence ministry spokesperson said its forces had dealt with drones which entered the country’s airspace early on Sunday, “in accordance with established procedures”, without attributing responsibility for the incursion.

Tensions have flared under the truce as both the US and Iran have sought to assert their control of the Hormuz strait. On 4 May, Donald Trump launched what he called “Project Freedom”, which was supposed to provide a route out of the Gulf for the hundreds of ships trapped by the war.

Iran, which closed the strait after the initial US-Israeli attack on 28 February, responded with attacks on US naval vessels, commercial vessels and oil facilities in the UAE, a close ally of the US and Israel.

Trump called off Project Freedom after 36 hours and the passage of just two US-flagged ships. Saudi Arabia had refused permission for US forces to use its bases and airspace for the operation.

Tehran has insisted that all ships passing through the strait coordinate with its armed forces and pay a $2m (£1.5m) toll. On Sunday, Iranian state media reported that a Panama-flagged vessel bound for Brazil had been allowed to sail through the strait.

Qatar denounced the strike on a ship in its territorial waters as a “serious escalation”. The country’s prime minister, Sheikh Mohammed bin Abdulrahman Al Thani, told Iran’s foreign minister, Abbas Araghchi, that using the strait as a means of exerting pressure would only deepen the crisis, and that freedom of maritime navigation should not be compromised.

Two critical issues which will be the heart of any future nuclear talks between the US and Iran are the suspension of Iranian uranium enrichment and the disposal of its stockpile of highly enriched uranium (HEU), which could be used to make about a dozen nuclear warheads. In an interview on Iranian state media late on Saturday, a military spokesperson said the country’s forces were at “full readiness” to protect the stockpile.

“We considered it possible that they might intend to steal it through infiltration operations or heliborne operations,” Brig Gen Akrami Nia said.

The UN nuclear watchdog, the IAEA, believes at least half the stockpile is deeply buried in Isfahan. Trump is reported to have been presented with military options for seizing the HEU, but the operation would have required a large number of troops and would have taken weeks.

Trump has said that the current ceasefire includes Lebanon, and has told Israel to stop the bombing of Hezbollah targets there. Israel has reduced the intensity of its campaign but has continued to carry out strikes.

Lebanon’s health ministry reported that 36 people had been killed and 74 wounded by Israeli strikes on Saturday. Among the casualties were several paramedics wounded in southern Lebanon. The Israeli military said, meanwhile, that it had intercepted Hezbollah drones approaching its troops in the area.