The bank announced earlier this week it was to implement a phasing out of a host of branches in a bid to save £100million.
The Barrhead branch in Arthurlie Street will close on April 29, 2020 followed by the Busby Road, Clarkston branch on July, 22, 2020.
It will join 81 other branches across the country to shut.
The three-year cost-cutting programme, announced by recently installed chief executive Debbie Crossland, is meant to simplify the business and refocus the bank on improved customer service.
The bank said this week that it would invest in self-service machines in branches, and in its flagship branches, which it said would be the "right locations to meet customer demand".
However, no roles are to be axed on the Island as the branch remains open.
It added that despite the closure programme, 65% of the population will still live within four miles of one of its branches.
The bank says it acquired a large network of branches from Lloyds Banking Group is 2013, with TSB holding twice as many branches per 10,000 customers than competitors.
"We realise this is hard news for our branch partners and will do everything to support those affected to offer voluntary redundancies and redeploy as many people as we can to other roles".
Customer banking director Robin Bulloch said: "We have made the hard decision to close some of our quieter branches and will fully support customers through this transition".