The bank's Church Stretton branch is due to close next May along with Ellesmere, in February, and Albrighton, in January. They are among 164 to be closed down as more customers use online services.
Mr Dunne said he would be writing to TSB chief executive Debbie Crosbie about the move.
“TSB made a virtue of developing its branch in Church Stretton to provide personal services for customers as other banks closed theirs in recent years," he said.
"It is therefore especially regrettable to learn that the bank has written to customers explaining it plans to close this branch. I have asked to meet the relevant decision makers in an attempt to persuade them to maintain the branch.
"I have a long championed the importance in maintaining a vibrant local economy of essential local services, including banking and cash handling for businesses and individuals alike.
"Local businesses will be particularly concerned about maintaining night safe deposit facilities, as well as cash and other face-to-face services for individuals in Church Stretton.
"I shall urge TSB to reverse its decision. TSB's nearest branch is 13 miles away in Shrewsbury, but even this is not open at weekends.”
Church Stretton's Shropshire councillor, Lee Chapman, said: “This is a real blow to the town and to the many customers of the bank, who will now have to travel miles away for face-to-face services.”
Shropshire councillor David Evans added: “This is another business gone from the High Street. Banks are an important business for small market towns. When we lost all three of our banks in Craven Arms the footfall to the shops fell.
"The bank we use only has branches in Bishop's Castle and Shrewsbury. All their branches closed everywhere.”
Mr Dunne was alerted to the closure by Ludlow town councillor Hilary Luff who is also campaigning for the branch to remain open.
Councillor Luff said: “My daughter opened her first bank account here as she was keen to be able to have personal advice for her banking needs. Having to travel to Shrewsbury makes this so much harder for her.
"It’s such a shame for the customers of this bank, many of whom are elderly and less likely to use mobile banking, that it will be closing,”