Theresa May has refused to say how she would vote if there was another EU referendum.
The prime minister, who backed Remain in last year's vote, was repeatedly asked if she would now vote for Brexit.
She told LBC radio: "I don't answer hypothetical questions."
The PM, who said during the general election campaign that the UK had a "brighter future" after Brexit, added: "I voted Remain for good reasons at the time but circumstances move on."
Downing Street sources suggested it would be ridiculous to say the prime minister's comments raise doubts about whether she will deliver Brexit, as some such as ex-UKIP leader Nigel Farage have said.
Conservative MP Bernard Jenkin, who was a leading campaigner for Brexit, said: "She is entirely right to avoid being divisive.
"She is seeking to unite the country, not to perpetuate referendum divisions."
By BBC Political Editor Laura Kuenssberg
Fairly or not, Theresa May's hesitation in giving her answer on this hypothetical question will give pause for thought to those who harbour suspicions of her real commitment to Brexit.
And her "open and honest" answer, which refused to come down on either side creates the strange situation where the prime minister appears unwilling to give full-throated support to her government's main policy.
Presenter Iain Dale told Mrs May that Health Secretary Jeremy Hunt had switched from Remain to Leave because former Chancellor George Osborne's gloomy economic predictions about the latter had failed to come true.
He asked Mrs May why she could not say she had changed her mind, given that she was leading the country into Brexit.
"Yes and I'm prime minister ensuring I'm going to deliver Brexit for the British people," she replied.
Pressed again, Mrs May said: "I could say I would still vote Remain or I would vote Leave just to give you an answer to that question.
"I am being open and honest with you. What I did last time round was I looked at everything and I came to a judgement and I would do exactly the same this time round.
"But we are not having another referendum and that's absolutely crucial."
Mrs May's second in command, First Secretary of State Damian Green, also refused to say whether he would back Brexit if there was a referendum now.
Mr Green, who was a board member of the campaign to keep Britain in the EU, told Channel 4 News: "I don't resile from anything I said during the election campaign."
But he added that it was a "meaningless" question and "purely hypothetical".
Liberal Democrat deputy leader Jo Swinson said: "It is staggering that even the prime minister isn't convinced by the government's approach to Brexit.
"If Theresa May doesn't have any faith in her own government's policies, why is she still driving this country towards the cliff edge?
"Theresa May says she would weigh up the evidence again, she shouldn't deny that right to the British people.
"The public must have the chance to change their mind if they want to, once the government comes back with a deal."
Former UKIP leader Nigel Farage tweeted: "How can Theresa May negotiate Brexit without believing in it?"
In the same LBC interview, Mrs May said she could not guarantee the status of the estimated 1.2 million UK nationals living in other EU countries if Britain leaves the bloc without a deal.
And she warned that rights held by more than three million EU nationals in the UK could "fall away" in a "no deal" scenario, something the government is actively preparing for if talks in Brussels fail.
"By definition, if there isn't a deal we won't have been able to agree with the EU what happens to UK citizens currently living in countries like Spain and Italy and other members of the EU," said the prime minister.
Labour leader Jeremy Corbyn tweeted: "Unacceptable. The Tories' chaotic handling of Brexit means no deal is a real risk. Theresa May must guarantee EU migrants' rights now."