More members of the public generally also want her to stay in place than think she should step down.
The results emerged amid more grumbling in Tory ranks about the progress she is making with Brexit and other policy.
Underlining the sense of danger she faces, there were also claims that more than a dozen donors to the Conservative Party had made clear at a fundraising event their desire for a change of leader – a move which could prove fatal if it gathers pace.
Mrs May, who left the UK today for an official visit to China, has been under intensifying pressure since her botched ministerial reshuffle.
A YouGov survey this week showed that 41 per cent of the public wanted Theresa May to stay put
The results of this poll clearly show that, on the whole, the public would rather see Theresa May remain as Prime Minister for the moment
She has been attacked for being “timid” in her European Union and domestic policies and Tories fear she is ready to consign Britain to “vassal” status during the planned transition period after next year’s official exit day.
At Westminster there are rumours that nearly enough no confidence letters have been submitted by Tory MPs to trigger a leadership contest while one source close to the European Research Group of strongly pro-Brexit Tories said the PM’s chances of survival were no better than “50:50”.
But today a YouGov survey this week of 1,669 people for political consultancy WPI Strategy found 41 per cent of the public think Mrs May should stay put, against 34 per cent thinking she should go and one in four voters undecided.
The figures were similar to those at the start of November.
The survey also found that potential rivals like Boris Johnson could harm the party’s chances
Significantly for Mrs May, support for her to remain PM is high among people who voted Tory in last year’s general election.
Of this group, 69 per cent thought she should continue as PM with only 18 per cent wanting her to stand down and 13 per cent unsure.
The survey also found that potential rivals could harm the party’s chances, with the public less rather than more likely to vote for it if it was led by Foreign Secretary Boris Johnson, Brexit Secretary David Davis or Home Secretary Amber Rudd.
People who voted Leave in the EU referendum are more supportive of Mrs May – 49 per cent want her to stay and 32 per cent to go – than Remain voters who are evenly split at about 37 per cent for each proposition.
Nick Faith, director of WPI Strategy, said: “The results of this poll clearly show that, on the whole, the public – and certainly those who voted for the Conservative at the last election – would rather see Theresa May remain as Prime Minister for the moment.
“However difficult it may be for an incumbent government to offer an agenda of renewal both domestically and on the international stage, that is the challenge facing the Prime Minister and her colleagues.”
The poll may steady some nerves but many in the party will be aghast at a claim that about a quarter of 50 Conservative donors at a fundraising event last week shouted in response to a Tory treasurer asking who could “possibly” want to replace the PM that they did.
“Among even the most loyal middle-ranking donors there is utter despair,” one unnamed businessman was quote saying.
Brexit Secretary David Davis could also harm the Conservative Party’s chances, said the survey
Conservative HQ said it did not comment on “private events”.
Former Tory donor Charlie Mullins, a keen Remain supporter, added his voice to calls on the PM to quit.
The businessman who founded the Pimlico Plumbers empire said: “I think it’s obvious now there is no future for her.
“She needs to go, the quicker the better.
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“We have all lost confidence in her and it is bad for business for her to linger on in there.
“We all know she is going to go. She may as well just book the removal lorry and move on.”
Mr Mullins also claimed there would be a second referendum on EU membership and that Britain would stay in the bloc.