Voting papers will be formally counted on Wednesday, but Sharon Graham, an assistant general secretary of the union, is expected to win.
It is believed she has achieved a few thousand more votes than Steve Turner, who had been tipped as the favourite candidate.
It is expected that the other candidate, Gerard Coyne will come third.
A spokesman for Ms Graham’s campaign said: “The sampling of the vote is going on. The more it goes on, the better it looks for us. Of course we will only get the result when the count is finished.
“But we are confident Sharon is going to win. If she does, it will be an historic victory for the campaign and the workers in Britain and Ireland.”
The official announcement of the ballot result had been due on Thursday but statements from the candidates confirming the votes are expected on Wednesday.
Ms Graham leads Unite’s Organising and Leverage Department, which specialises in taking on hostile employers.
She has led recent disputes at British Airways and Crossrail as well as campaigning to unionise Amazon.
She described herself as the “workers’ candidate”, pledging to take Unite “back to the workplace”.
She has said she wants to rebuild the union as a movement which delivers industrially and politically.
Speaking about the election campaign, she said: “My slogan all along has been ‘Back to the workplace’ to build the union to fight for jobs, wages and conditions.
“That has been rewarded with huge support for me in big industrial branches like Hinkley Point, London construction, Ireland construction and Vauxhall.
“My campaign co-ordinators estimate that my branch nominations cover a potential voting base of 250,000 members – and that’s before the 300,000 women in the union have had their say.
“I am not a member of any Unite or Labour faction – other than my own supporters group.
“Unite members at the workplace want real change, not a settling of old scores or a Westminster rematch.”