RBS branches in Bellshill and Tannochside are at risk.
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A total of 62 branches ranging from the Isle of Skye to Gretna are currently under threat, with the potential loss of 158 jobs.
The banking group, which also operated the NatWest brand across the UK, is closing 259 branches in total, although the 197 NatWest operations are restricted to England at present.
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Hide AdUnite regional officer Lyn Turner said: “RBS’s 2016 annual report states that they want ‘to transform the bank into the number one for customer service, trust and advocacy’.
“What we have in 2017 is a plan which amounts to institutional financial vandalism on a scale which has never been seen in this country ever before.
“Customer service? Trust? Don’t make me laugh.”
An RBS spokeswoman said: “More and more of our customers are choosing to do their everyday banking online or on mobile.
“Since 2012 the number of customers using our branches in Scotland has fallen by 44%.
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Hide Ad“Only 1 per cent of our customers in Scotland now use a branch regularly while the number of regular mobile users has increased by 39 per cent since 2015.
The spokeswoman added: “We expect these closures to result in around 158 redundancies. We realise this is difficult news for our colleagues and we are doing everything we can to support those affected.
“We will ensure compulsory redundancies are kept to an absolute minimum.”