Union members are demanding a 6 per cent pay increase and the scrapping of performance-related pay.
A series of selective strikes at a number of TSB's larger regional offices will start within a month as a result of the vote, the union said.
John Townsend, Bifu's assistant secretary, said: 'The union has rejected a 2 per cent pay package and opposes plans by TSB to put staff on to performance-related pay.
'Performance pay is there to cut costs, not to improve service to the customer. Our members strongly oppose it.'
Bifu said yesterday that its members voted 3,073 to 2,980 in favour of strike action. There was a 41 per cent turnout in the ballot, which has been running since the beginning of the month.
Bifu claims 90 per cent membership of TSB's 16,000- strong branch workforce. That makes it one of the most heavily unionised of the high street banks, alongside Midland Bank.
TSB made no formal comment last night as its industrial relations department was considering the implications of the vote.
But sources close to the bank wondered whether Bifu would think it had a strong enough mandate for strike action with only 20 per cent of the TSB membership voting in favour.
Bifu managed to shut more than 400 TSB branches in a previous one-day strike in January 1993 over compulsory redundancies, a confrontation that the union claimed as a victory.
Yesterday's result goes next week to Bifu's executive council, which will formally decide on the exact form of action, a spokeswoman said last night.
The wording on the ballot paper called for 'a series of selective strikes', which are likely to be one-day stoppages in a number of large regional centres.
The vote followed a consultative ballot earlier in the year in which 80 per cent of the TSB membership rejected the 2 per cent pay package.
There has been widespread consternation among banking union members over an industry-wide move away from across-the-board pay increases towards performance- related pay.
That, coupled with heavy job cutting, has been prompted by the banks' stagnant operating profits, which have made cost-cutting the management's top priority.
In July TSB said it was making 150 senior managers redundant, while Bifu said another 700 clerical jobs were at risk. Falling bad debts enabled TSB's pre-tax profits to leap from pounds 80m to pounds 226m in the six months to 30 April.Reuse content