The YouGov poll for The Times of 1,400 eligible voters put Mr Corbyn on 53%, 32 points ahead of Andy Burnham.
The left-wing MP said his campaign was "going very well" but ballot papers had not yet been sent out.
Alastair Campbell has warned Labour risks "driving itself off a cliff" if Mr Corbyn is elected.
Tony Blair's former spokesman is urging support for "anyone but Corbyn" in the leadership vote, arguing that the Islington North MP could never win a general election.
Mr Corbyn, a backbencher who was seen as a rank outsider at the start of the campaign, has been packing out meeting halls across the UK, with many of his supporters joining the Labour Party to vote for him.
The party has gained 79,000 members since the general election and 145,000 registered supporters.
If the findings of the YouGov poll were repeated in the leadership election, Mr Corbyn would win without the need for second preferences to be counted.
YouGov president Peter Kellner said he "would personally be astonished if Corbyn does not end up as Labour's leader" despite voting not starting until Friday and the result not being declared until 12 September.
But others were warning about reading too much into one poll, given the difficulties of carrying out a representative survey of Labour's membership and those who have signed up as party supporters.
Joe Twyman, also from YouGov, cautioned that the poll was a snapshot and there was still "a lot to play for" for all of the candidates in the race.
"The ballot papers haven't gone out, we've still got a month to go. What it shows to me is that there's a lot of movement on our previous poll and there still could be a lot of movement in the remaining four weeks," he told BBC Radio 4's World at One.
A spokeswoman for Yvette Cooper's campaign said: "This does not reflect our extensive phone banking data, which does not suggest any single candidate will receive 50% of first preferences."
Mr Corbyn told BBC Radio London his campaign was going "very well" but added: "We should be a little bit cautious."
"There is still time for people to register as supporters and no ballot papers have been sent out yet", the Islington North MP added.
Asked about Mr Campbell's criticism, he said the leadership contest was "very democratic".
He said the large number of new Labour Party members was a "good thing", adding: "If they choose somebody, the leader that others don't like, well I think we have to accept the democratic process."
But he described "Corbynmania" as a "bit embarrassing".
Former Labour Welsh Secretary Peter Hain told the World at One he did not think Mr Corbyn would make a successful party leader, or lead Labour to victory in 2020 "unless there is some kind of major seismic change in British politics".
But he said the choice of candidates on offer was "underwhelming", saying that Andy Burnham and Yvette Cooper,  who he described as "the most credible candidates", had failed to capture "the mood in the party".
At-a-glance profiles of the four contenders
Mr Campbell said on his blog that he thought Mr Corbyn was "an OK guy, a good MP, and his stance clearly chimes with many people's views of anti-austerity" but his ability to lead and hold the party together "is likely to be low".
He added: "Once the pressures of real, difficult decisions and the day to day leadership of the main opposition kick in, I fear that activists currently cashing in on perceived 'betrayal' by past Labour leaders are going to end up feeling very badly let down."
Mr Campbell was criticised by left wing Labour MP and former leadership hopeful Diane Abbott, with the two clashing on Twitter about Labour's record.
On Monday Barry Sheerman became the latest Labour MP to call for the leadership contest to be "paused" over fears it had been infiltrated by supporters of other parties.
These fears have been rejected by Labour, which says robust checks are in place.
Jeremy Corbyn has urged his supporters to be "a bit cautious" after an opinion poll suggested he will romp to victory in Labour's leadership contest.