Around 40 Scottish branches are being axed, including those at Mintlaw, Fraserburgh and Banff, and in the region of 200 jobs are under threat.
The bank says the move has been prompted by the trend towards electronic banking - but there are warnings the closure programme will have a massive impact on local communities.
North-east politicians have called for an immediate rethink and some are seeking an urgent meeting with the Clydesdale’s chief executive David Duffy.
SNP MP Dr Eilidh Whiteford, who represents Banff and Buchan where three branches are set to close, said: “Closing bank branches in rural towns is extremely damaging to communities.
“Not only are they used by my constituents, but these branches help to support a number of local businesses who will lose vital local banking services.
“The closure in Mintlaw is also worrying given it is the last bank to serve the village.”
Gillian Martin, SNP MSP for Aberdeenshire East, and Central Buchan councillor Jim Ingram have both written to the bank urging bosses to reconsider their decision to shut the Mintlaw branch.
Ms Martin, a member of Holyrood’s Economy, Skills and Fair Work Committee, said: “I know many local businesses rely on their High Street branch and the ability to bank cash takings at the end of the business day.
“I am also concerned that the closure of the branch is going to hit the most vulnerable in the community, particularly the elderly who may not have access to the internet, or need face-to-face assistance for their banking transactions.”
Councillor Ingram said: “I have urged the bank to reconsider its decision based on the essential service the branch provides to the communities in and around Mintlaw.
“If retaining the branch as it is would not be possible, I have asked the bank to consider other options, such as part-time opening or keeping a cash machine service, that would maintain a presence in the town.”
In a statement, Clydesdale Bank said its first priority was to customers and it would be “working extensively” with all affected to ensure the transition to their new branch would be “as smooth and sensitive as possible”.