Senior Labour sources say they understand Mr Deans will not stand when the post is up for re-election later this month.
There had been speculation that Mr Deans might be ousted at a meeting of the local Labour party today.
But his position was not discussed at the meeting this afternoon, which Mr Deans did not attend.
However, a source said Labour "understood" that Mr Deans would not be putting his name forward for re-election at the local party's Annual General Meeting (AGM) in three weeks time.
The local party has been at the centre of a media firestorm for months amid allegations Unite attempted to rig the contest for Labour's next Westminster candidate for Falkirk.
The current MP, former Labour politician Eric Joyce, has announced he is standing down at the next General Election, due in 2015.
Joyce was expelled from the Labour Party after brawling in a Commons bar.
Mr Deans helped recruit a large number of new local party members, many who worked at Ineos, in a bid to have Unite's preferred candidate, Karie Murphy, selected to fight the Falkirk seat at the next election.
Both were suspended by Labour over the move but reinstated after witnesses withdrew evidence during an inquiry.
Mr Deans quit his job at the Ineos plant at Grangemouth on Monday. He was alleged to have used work time for local Labour Party business rather than legitimate trade union activity in his role as Unite convener at the plant.
A separate Ineos probe into Mr Deans's activities sparked a wider dispute over the firm's plan to freeze wages and cut pensions. Unite backed down after Ineos threatened to close the petrochemical side of the plant.
Yesterday Scottish Labour leader Johann Lamont, who critics have accused of being reluctant to speak out about the Falkirk vote-fixing row, branded those involved as arrogant.
She said: "What happened at Falkirk was completely unacceptable.
"We acted quickly to expel Eric Joyce from the party after his behaviour was exposed.
"But the process to select a new candidate was abused by people with personal agendas who abused the power they had.
"It may have been bruising and taken longer than I wanted but we acted and the process to pick a Scottish Labour candidate to offer to the people of Falkirk has begun again.
"I am determined that this time it will be an open, free and fair contest.
"But of all the abuses which happened in the past, I believe the greatest one is this. That there were people within the party and, indeed, outwith it who believed the selection was the most important thing.
"That whoever was chosen would be accepted by the people of Falkirk. That arrogance is unforgivable.
"The people of Falkirk must be at the heart of this. We need to find someone who can truly represent them and who they can trust."