The five branches, which employ 49 people, may be moved as the Post Office seeks partners for the controversial venture.
The other three Scottish branches affected are in Perth, Alloa and Stornoway.
The Post Office, which could work with the likes of WH Smith, the Co-operative or other businesses, said the range of services would not be affected and branches could be open longer.
But there was dismay in the communities affected and from the Communication Workers Union (CWU), whose members are among the 49 staff affected.
It also raised fears of longer queues after problems at other franchised post offices.
Gregg McClymont, Labour MP for Cumbernauld, Kilsyth and Kirkintilloch East and Shadow Pensions Minister, said: "I am seeking an urgent meeting with Post Office Ltd to discuss what this means for local people."
Labour's shadow minister for Postal Affairs, Ian Murray MP, said: "People across Scotland really value their post offices and want to be able to access a range of services at them.
"The announcement today raises huge concerns that certain services, like motor vehicle licences, passport check and send services, bureau de change and travel insurance on demand will no longer be easily available to local people."
The MP for Edinburgh South added: "The UK Government claims to have no planned closure programme, but this would seem to be closure and reduction of services by stealth.
"They must answer how the future of our cherished post offices is to be protected."
The CWU is seeking guarantees because rules over the transfer of employees' rights – known as Tupe – were not applied when the last tranche of Crown Post Offices were franchised.
General secretary Billy Hayes said a Consumer Focus report had criticised the performance of post offices franchised to WHSmith, saying they were the worst on queue times and scored badly on quality of service.
The change relates to Crown Post Offices.
Across the UK, 70 Crown Post Offices have been selected for franchising.