Among branches earmarked for closure are Aberdeen Riverside, Bathgate, Bearsden, Dalkeith, Edinburgh University, Fraserburgh, Grangemouth and Rosyth.
Parent company CYBG will also shut a number of Yorkshire Bank branches, bringing the total number of closures across the UK to 79.
The Unite union has said the decision will be “deeply devastating” to the employees.
It is the largest-ever closure plan by the company and will radically change the face of the bank, according to Unite.
National officer Rob MacGregor said: “Staff across the Clydesdale and Yorkshire Bank will be devastated to learn that 79 branches will close and consequently over 400 colleagues will lose their jobs.
“Unite is clear that 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 that they will reconsider closing any bank branches that are the last bank in a town.
“Unite workplace representatives will be able to assist members over the coming day and weeks to support staff impacted by this announcement.”
Clydesdale said that, since 2011, the number of customers using their bank for day-to-day transactions had fallen by a third.
A statement said: “This ongoing decline in branch usage, married to a sharp and sustained increase in digital and mobile engagement, has driven a shift to providing greater access to day-to-day banking services remotely, on the move and outside of normal business hours.
“The bank continues to reshape its service offering in response to these changing needs. To support this transformation the bank is committing £350 million 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. However, there will inevitably be employee number reductions as a result of these changes.
“We anticipate that around 200 Clydesdale Bank staff will be at risk of redundancy and subject to our redeployment process. All employees affected by these changes will be fully supported through the process.”
Separately, about 200 Yorkshire Bank staff will be at risk of redundancy.
Gavin Opperman, customer banking director at Clydesdale Bank, said: “While the decision to close any branch is never an easy one, it is important that we, in line with other banks operating in the UK market, continue to respond to changes in the way customers want to bank with us.”
Full list of Clydesdale branches set to close
East Kilbride, Stuart St
Edinburgh Leith Walk
Glasgow, Aikenhead Road
Glasgow, Bath Street
Glasgow, Charing Cross