Theresa May will say the Iran nuclear deal is “vitally important” in her first major intervention on the issue since the election of Donald Trump, who has said he plans to tear up the agreement secured by the Obama administration.

Speaking on Wednesday, the prime minister will tell Gulf leaders in Bahrain that Iran remains a threat to the stability of the Middle East and the UK would not ignore “aggressive” Iranian actions in Syria and Yemen.

Addressing the plenary of the Gulf Cooperation Council (GCC) in Manama, May will say she is realistic about the threat Iran poses in the Gulf and the wider region, and prepared to “confront state actors whose influence fuels instability”.

However, the prime minister will also underline the importance of the 2015 agreement, saying it makes more direct engagement with Iran on issues of concern possible in a “clear-eyed, hard-headed manner”.

“We secured a deal that has neutralised the possibility of Iran acquiring nuclear weapons for more than a decade,” she will say. “It has already seen Iran remove 13,000 centrifuges, together with associated infrastructure, and eliminate its stock of 20%-enriched uranium.

“That was vitally important for regional security. But we must also work together to push back against Iran’s aggressive regional actions, whether in Lebanon, Iraq, Yemen, Syria or in the Gulf itself.”

Trump has threatened to scrap the deal, signed by six major powers including Britain, in which Iran agreed to scale down its nuclear programme in exchange for sanctions relief. US Republicans opposed the deal brokered by the Obama administration and called for sanctions to be increased.

In March, Trump told supporters that his number one foreign policy objective was to “dismantle the disastrous deal with Iran”, which he has since called “the worst deal ever negotiated”.

May’s comments make it clear to the US president-elect and the Iranian government that the UK would consider itself bound by the deal even if the US withdrew. Britain, along with the other signatories France, Germany, Russia and China, has resumed trade relations with Iran since sanctions were lifted.

On Tuesday, Iran’s president, Hassan Rouhani, said Trump could not unilaterally dismantle the deal. ”Do you think he can rip up the [deal]?” he said. “Do you think we and our nation will let him do that?”

The outgoing director of the CIA, John Brennan, has told of disastrous consequences if Trump renounces the agreement, warning that this would be the “height of folly” and any breakdown in relations could lead to a nuclear arms race with other Gulf states.

Israel, which opposes the deal, has said the terms do not prevent Iran from acquiring a nuclear weapon within the next decade. The Israeli prime minister, Benjamin Netanyahu, has said he intends to press Trump to renege on the “bad deal”.

May, who is holding bilateral meetings with six Gulf leaders during her two-day visit to Bahrain, will lead a focused session on Iran and regional security issues after the speech.

Downing Street said she would highlight key areas of concern, including the al-Quds force of Iran’s Revolutionary Guards acting in Syria to shore up the Assad regime and working with Hezbollah in Lebanon.

Although her remarks on Wednesday are likely to be interpreted as a warning to Trump not to pull the US out of the deal, May told reporters she got on well with Trump on a personal level in their two phone calls, saying he was “very easy to talk to”.

“He is someone who very much values the relationship he has with the UK,” she said.

“Whenever we’ve talked, that’s been one of the key things we’ve talked about; the depth of our special relationship and the fact we want to maintain that and build on it for the future.”

On Tuesday, May met the leaders of the United Arab Emirates, Kuwait and Saudi Arabia for bilateral talks lasting roughly 30 minutes each, and dined with all six GCC leaders on Tuesday night. Downing Street sources said talks with the UAE focused on trade and investment.

Saudi Arabia’s role in the Yemen conflict was raised with King Salman, with the need for a comprehensive Saudi investigation into human rights abuses emphasised during the meeting, a source said. May will also criticise Iran’s support for Houthi rebels in the conflict in her speech on Wednesday.

“The PM said it was very important lessons are learned and acted upon,” the source said, but Downing Street did not call for an independent UN investigation into Saudi forces’ conduct in Yemen.

More than 6,800 people have been killed in the country, the majority in Saudi-led airstrikes in support of the Yemeni president, Abd-Rabbu Mansour Hadi.

Downing Street would not confirm whether specific cases of imprisoned or exiled dissidents in the Gulf had been raised during the talks, but she is due to have a working lunch with the king of Bahrain on Wednesday.

MPs from the all-party parliamentary group on democracy and human rights in the Gulf wrote to May on Tuesday, asking her to raise the case of Nabeel Rajab, the president of the Bahrain Centre for Human Rights, who is facing up to 15 years in prison for offences including insulting the state in an article he wrote for the New York Times.

“The European parliament, the United Nations and the US state department have all expressly called for his release from prison,” the letter said. “To date, the United Kingdom has not done likewise.”