Tony Blair has hit back at Theresa May after she accused him of insulting the office of prime minister by advocating a second Brexit referendum to Britain’s negotiating partners.

In a strongly worded rebuke, May said on Saturday: “For Tony Blair to go to Brussels and seek to undermine our negotiations by advocating for a second referendum is an insult to the office he once held and the people he once served.”

Downing Street has been irritated by what it regards as out-of-touch grandees using their formidable contacts books to pursue the goal of keeping Britain in the EU while May has been negotiating an exit deal.

But Blair’s office published a statement on Sunday saying it was “not irresponsible or insulting to put forward an alternative way to achieve resolution” since it was “perfectly clear neither the British people nor their parliament will unite behind the prime minister’s deal”.

Blair added: “I have always said, and did again in my speech on Friday in London, that I personally sympathise with the PM’s heavy burden in doing her job. I do not disrespect her at all. I understand her frustration. But I profoundly believe that the course she is pursuing will not work and is emphatically not in the national interest. And that’s the reason I am speaking out and shall continue to do so.”

Former prime ministers have generally restrained themselves from intervening in frontline politics in recent years, and the public spat between the pair underlines the bitter tone of much of the Brexit debate.

Blair called for MPs to be given a series of votes on alternative Brexit options – an approach now favoured by several cabinet ministers, including Amber Rudd and Liam Fox.

“The sensible thing is now to allow parliament to vote on each of the forms of Brexit canvassed, including the prime minister’s deal. If they can’t reach agreement then the logical thing is to go back to the people,” Blair said.

Allies of the former Labour prime minister pointed out that May had not levelled a similar charge at the former Tory prime minister John Major.

Together with Nick Clegg before he took a senior job at Facebook, and Peter Mandelson, Blair has been urging senior EU27 politicians to keep open the option of a second referendum.

He suggested on BBC Radio 4’s Today programme on Friday that private conversations had convinced him other EU states might be willing to offer concessions on free movement to help clinch a referendum win for remain. “As one senior European leader said to me just a short time ago, let’s be clear, in the last 30 months European politics has been transformed,” he said.

By contrast, May was keen to underline her objections to the idea of a second referendum, which has been gathering momentum at Westminster and which reports have suggested her own aides are considering.

“We cannot, as he would, abdicate responsibility for this decision. Parliament has a democratic duty to deliver what the British people voted for,” May said. “I remain determined to see that happen. I will not let the British people down.”