The deadline for a ballot of Fujitsu workers in Manchester has been extended to 1 September, which means that if there are enough votes, a strike could hit the IT company late next month.
The proposed strike action is over the sacking of Crewe-based union rep Alan Jenney, whose case will be heard at an employment tribunal in Stoke on 25 August.
The move could impact a number of Fujitsu's contracts, including with the Home Office, HM Revenue & Customs, Marks & Spencer and the Post Office.
The Unite union claims that Fujitsu has "singled out" Jenney, who worked at the company's Crewe office, and made him redundant, whereas it has helped to redeploy other staff where possible.
It also said that Fujitsu had breached company processes and agreements in the way it had dealt with Jenney's situation. Jenney had worked as an IT professional for Fujitsu for 17 years.
Unite is currently balloting more than 300 members at Fujitsu's Manchester site on taking strike action or action short of a strike over the issues. The ballot was originally due to close on 5 August, but was extended due to "technical issues".
If the workers vote for a strike, it would be further compounded by a current ballot being pursued by members of the Public and Commercial Services (PCS) union for strike action over Fujitsu's "insufficient" pay rises.
In June, the employees at Fujistu's Crewe office carried out a one-day strike, began to work to rule and instituted a policy of non-cooperation from 30 June. The employees are continuing to work to rule.
At the employment tribunal next week, Unite is supporting an interim relief claim, which, if successful, would mean that Fujitsu would be forced to re-employ Jenney until a full hearing is held.
Meanwhile, Unite said it was also continuing talks with Fujitsu's management team at conciliation service ACAS.
Kevin O'Gallagher, Unite national officer for IT and communications, said: "We are behind Alan Jenney 100 percent and this dispute will spread throughout Fujitsu's UK operations until justice is done.
"The attack on Unite's reps must cease immediately and there must be a marked improvement in the company's employee relations strategy."