The independent savings bank will close its Motherwell and Baillieston branches, and part-time outlets in Muirhead and Shotts, on 28 August.
The bank said the move was a result of changes in branch usage and unsustainable operating costs.
Its core branches in Airdrie, Coatbridge, Bellshill and Falkirk will continue to operate as normal.
ASB said customers of the four closing branches could continue to access services online and by telephone.
About 20 jobs are expected to be lost as a result of the closures, although ASB said six new posts would be created following the strategic review.
The review covered all aspects of the 180-year-old bank’s high street network and online and telephone operations.
In addition to the branch closures, ASB plans to “streamline and modernise” back-office processes, restructure departments and review staff skills.
ASB chief executive Rod Ashley said: “ASB operated for many years as a traditional branch-based bank.
“More recently, we have modernised our service delivery by offering online banking through our website and telephone banking through our customer contact centre.
“These improvements have proven extremely popular by providing customers with easy access to our services, when and how they choose.
“However, they have required significant and ongoing investment at a time when the banking sector has had to address many financial and operational challenges, ranging from the increased costs of stronger regulatory requirements to higher transaction-processing costs.
“As a result, the board has concluded that for Airdrie Savings Bank to grow and expand its services to the communities it serves in the future, we need to close some branches that are unsustainable, due either to low customer footfall and demand or the cost of maintaining them to a satisfactory standard.”