Leicester College, which is based in the city, has given staff until the end of January to take up the offer of a pound;1,800 one-off payment.
Accepting the cash will mean that lecturers must sign a contract that will cut their holiday entitlement by four days a year.
Natfhe, the lecturers' union, is to ballot its members at the 42,000-student college on possible indefinite strike action over the contract.
"Asking us to lose four days' holiday is appalling," said branch secretary and graphic design lecturer Kate Drew. "They should be improving conditions for lecturers, not eroding them.
"Most people have turned down this pound;1,800 bribe."
Another lecturer, Dave Hornsby, who is also branch chairman, said that staff had reached the end of their tethers and described the offer as a cheap bribe.
"People are very upset," he said, "and we can't rule out the possibility of an indefinite strike."
Natfhe says it is also concerned about the working conditions for part-time staff at Leicester College, who are paid an hourly rate. The union claims that many of them are having to wait for more than two months to get their money.
Bosses at the college say that the one-off payment was offered because the union had withdrawn from talks over a contract with reduced holidays.
"The contract they have at the moment is 48 days' holiday, which is extremely difficult because it does not leave us enough time for staff development," said a college spokeswoman.