Corbyn received just shy of 60% of the vote when he was elected to the job in September 2015, reports the Press Association.
The new poll is likely to give Corbyn’s campaign an injection of vigour as the fight for votes continues although he still faces challenges among the wider UK population.
A survey conducted earlier this month found he is as unpopular as Nigel Farage amongst the general public, giving both men a -28 net rating.
The latest poll will do little to assuage the concerns of the 172 Labour MPs who backed a no-confidence motion in Corbyn - a move which ultimately led to Mr Smith’s challenge.
It puts Corbyn ahead in all three groups of people who are eligible to take part in the vote.
The poll found that 52% of full members intend to back Corbyn compared to 40% for Smith.
Meanwhile, among registered supporters it was 70% to 25% in favour of Mr Corbyn.
Corbyn also has the backing of the union affiliates by 54% to 33%.
An estimated 640,000 members and supporters have been tasked with choosing between the pair.
Voting in the contest is already under way with a winner to be announced on September 24.
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