The Bearsden branch of the Clydesdale Bank is closing on May 2 as part of a company wide restructure.
A total of 79 branches of The Clydesdale and Yorkshire banking group (CYBG) are closing with the loss of 400 jobs.
The Unite union said this is the largest-ever closure plan by the bank.
Unite national officer Rob MacGregor said local communities would be left with less choice in banking services.
Mr MacGregor said: “Staff will be devastated - over 400 colleagues will lose their jobs.
“The closure of a third of the bank’s branches will not only be concerning for staff but the local communities which will see their bank branch close. This cost-cutting plan leaves customers with less choice for local banking.
“The union has called on the bank to give a commitment to mitigate compulsory redundancies where possible and reconsider closing any bank branches that are the last bank in a town.”
He added: “Unite workplace representatives will assist members over the coming days and weeks to support staff impacted by this announcement.”
Clydesdale said the move had been prompted by the trend towards electronic banking.
It said the banking industry in the UK had seen bank visits fall by a third since 2011.
A statement from the bank said: “The Bank continues to reshape its service in response to these changing needs.
“To support this transformation the Bank is committing £350m over the next two years, embracing digital innovation while continuing to invest in a more sustainable branch network to deliver a superior customer experience.
“The Bank’s first priority is to our customers and we will be working extensively with impacted customers, local communities and relevant stakeholders to ensure that the transition to their new branch is as smooth and as sensitive as possible, particularly where vulnerable customers are concerned.
“It is also the Bank’s intention to try to find roles for frontline branch staff either within other branches or elsewhere in the Bank, wherever possible.”
Clydesdale customer banking director Gavin Opperman added: “While the decision to close any branch is never an easy one, it is important that we, in line with other banks in the UK market, continue to respond to changes in the way customers want to bank with us.
“The changes announced today continue our journey towards a model that combines an enhanced digital platform with a right sized branch network; allowing customers to interact with us through a wide range of channels - mobile, online, telephone, in-branch - whenever and however they want.”