The GMB union says it has enrolled a majority of workers at Amazon's Coventry warehouse which qualifies them for recognition by law.
It has written to the company asking to be recognised.
Amazon says it "respects its employees' rights to choose to join or not join a labour union".
The GMB believes it is on the cusp of a historic victory after a decade of trying.
If successful, it would mean Amazon would have to negotiate with workers about their pay, holidays and sick pay.
Amanda Gearing, senior organiser of the GMB, said that the process of establishing a union was "never straightforward".
She said: "There is a full process in place to try and prevent the GMB from forming, but we have the numbers now and Amazon will go out of their way to flood that warehouse with more workers so the numbers are different."
Amazon has 10 days to respond.
It said it regularly reviews pay, and that starting pay was between £11 and £12 per hour.
"Over the past seven months, our minimum pay has risen by 10% and by more than 37% since 2018," it added.
The union estimates that there are 1,300 workers at the Amazon distribution centre in Coventry. It says a majority - nearly 700 - have joined the GMB and says that means it has met the threshold for statutory recognition.
Darren Westwood, who works at the warehouse, and who has been at the forefront of getting people to join the union, says it is "fantastic" that recognition could be happening soon.
He said: "It's just so exciting because we've taken on one of the biggest companies in the world and won."
If Amazon does not grant recognition, the body responsible for resolving recognition disputes, the Central Arbitration Committee, could be asked to step in.
The CAC could automatically grant recognition if it is persuaded that a majority of the workforce wants the union to represent them. The workforce could be required to vote to show its support for this.
Workers at the Coventry warehouse first started protesting about their pay last August - when only 30 of them were members of the GMB. They held the first ever Amazon strike in the UK in January.
Since then, the company has increased its minimum starting wage to between £11 and £12 an hour, depending on location. The union is calling for an hourly wage of £15 an hour.
But the dispute has always been about more than money. Mr Westwood said a union was needed because, "it sometimes feels as if the management has no humanity".
Having a union, he said, was "about having that person on your side. It's about having protection in your back pocket."