Forty per cent of voters are confident Labour would handle the health service well - almost identical to the 38 per cent who think the same of the Tories.
It is a marked contrast to previous election campaigns when Labour has enjoyed a double-digit lead over their political rivals on the topic.
The finding is a major challenge for Jeremy Corbyn as he puts his party's record on the health service at the centre of his bid to win power.
Tory figures said the finding suggested the Labour leader had been robbed of “the one thing left in his locker” for the campaign battle to come.
An exclusive poll of almost 2,000 voters by ORB International conducted after the snap election was called shows voters' views on the Tories and Labour.
Respondents were asked to say how much confidence they had in the two parties when it comes to a string of key issues facing Britain.
The Conservatives were found to have major leads on the economy, handling immigration, delivering Brexit and defence.
In each case around twice as many voters had confidence in the Tories handling the issue than Labour.
However the most surprising finding is on the NHS, where Labour’s traditional lead in trustworthiness appears to have evaporated.
Polling experts put the change down to Mr Corbyn's perceived weak leadership in the eyes of voters having an impact on Labour’s overall brand.
In the 2015 general election Labour put the NHS front and centre as pollssuggested it was one of the few areas they were more trusted on than the Tories.
Giles Kenningham, the Tory director of communications during that campaign, said: “Labour has traditionally owned the issue of the NHS - they were almost untouchable on it.
"For the Conservatives the best they could ever do was neutralise the issue and move the debate on to being the party of economic competence.
"With Corbyn struggling to carve out a distinctive message , the NHS looked like the one thing left in his locker he could default to.
"Now it begs the question: Where does Labour go from here? They've got a shopping list of policies but no overarching strategy and more importantly no clear vision to sell the country."
Other findings in the poll highlight the challenge Labour faces if it is to win back power for the first time since 2010.
Mrs May is viewed more favourably among working class voters and public sector employees than Mr Corbyn, despite those demographics traditionally voting Labour.
She has also got strong support among Tory voters, with just one in 20 Conservatives having an unfavourable view of their leader.
For Mr Corbyn, that figure is much higher. One in four Labour supporters - 23 per cent - have an unfavourable view of him.
For Paul Nuttall, the Ukip leader whose party has undergone a series of damaging splits in recent months, 23 per cent of his party’s supporters have concerns. For Tim Farron, the Liberal Democrat leader, that figure is 12 per cent.
Johnny Heald, Managing Director at ORB International, said: “As the campaign starts, Jeremy Corbyn has a significant challenge ahead if he is to convince people that he in the right man to take care of the economy, handle immigration, deliver a Brexit deal that is good for the UK and manage our defence. Almost one in two adults report to be 'not at all confident' in his ability to deal with these issues.”
Politicians on all sides have cautioned against reading too much into the pollsafter they failed to accurately predict the outcome of the 2015 election.
Tory figures have been warning they may be over-exaggerating support for the Conservatives after experts failed to sport the party’s majority that year.
They are also warning activists not to be complacent despite the party’s 20-point lead over Labour according to recent surveys.