Yvette Cooper has rejected Tony Blair’s assertion that the UK should have supported Donald Trump’s initial airstrikes on Iran, saying Britain had to “learn the lessons” of mistakes made in Iraq.

At a private lunch event on Friday, the former Labour prime minister said Keir Starmer “should have backed America from the very beginning” and let the Trump administration use British airbases, adding: “If they are your ally and they are an indispensable cornerstone for your security … you had better show up when they want you to.”

Blair’s intervention comes as Trump intensifies his criticism of Starmer over the lack of immediate UK support for the US-Israeli strikes on Iran, saying on social media: “We don’t need people that join Wars after we’ve already won!” Asked about Blair’s comments, Cooper told Sunday Morning with Trevor Phillips on Sky News: “I just disagree.”

The foreign secretary added: “There are some people in politics who think that we should always agree with the US whatever. There are other people in politics who think we should never take action with the US again whatever the circumstances. I don’t think either of those positions is in the UK national interest, and it is the responsibility for Keir Starmer to act in the UK’s national interest for British citizens.”

Asked if she was calling Blair “a poodle”, she said: “I think the point is to make sure that, actually, we learn the lessons from some of the things that went wrong in Iraq, and I think that is exactly what Keir Starmer has done.”

Blair, who has been fiercely criticised over the past two decades for his decision to join the US invasion of Iraq in 2003, made the comments at a private Jewish News event on Friday, adding that he had already made his criticism clear to the government. He told those at the event that alliances were tested “when it’s hard”, saying the bases were needed for refuelling and adding that the conflict was “not like Vietnam”, the Mail on Sunday reported.

“It’s not like the Iraq campaign [where] we sent thousands of British troops. So I just think you’ve got to make the argument to the public about this,” he said. “You’ve got to say to them the American relationship matters. It matters particularly today. It’s not a question of whether it’s this president or that president.”

A spokesperson for Blair said the comments came at a private event and were not intended to be made public.

Responding to repeated stinging criticism of Starmer by Trump in the past week – which has included the US president saying on Saturday that Starmer’s help was not needed in Iran, and earlier in the week that the prime minister was “not Winston Churchill” – Cooper said the UK had to “take decisions ourselves” and act in its own national interest.

She told the BBC’s Sunday With Laura Kuenssberg: “It’s for the US president to decide what he thinks is in the US national interest, and that’s for him to do.

“But it is our job as the UK government to decide what’s in the UK national interest, and that doesn’t mean simply agreeing with other countries or outsourcing our foreign policy to other countries.”

Her comments came as the Iranian ambassador in London warned the UK to be “very careful” about becoming further involved in the war. The UK has given permission for the US to use British bases for “defensive strikes” on Iranian facilities, but has not taken part in any direct attacks.

Seyed Ali Mousavi told Sunday with Laura Kuenssberg his country would have a “right to self-defence” if the UK directly joined US-Israeli attacks on Iran. He warned that the British government, and other nations should be “very delicate, very careful” in their actions, adding he believed the British government had learned lessons from the 2003 invasion of Iraq.