RBS has told staff that up to 12 offices could close in England, while Scotland will also be badly hit.
A third of the job losses are as a result of RBS selling 318 of its branches to Santander.
RBS announced about 4,000 job cuts last year as it it tries to restructure its beleaguered business.
RBS had been told by the European Commission to reduce its number of branches in order to safeguard competition concerns after it was bailed out by the UK government.
The jobs to be slashed include 1,000 in IT support and 2,500 in support services to its other businesses.
RBS announced about 4,000 job cuts last year as it restructures its business.
Rob MacGregor, Unite national officer said: "The news that The Royal Bank of Scotland is to cut another 3,500 staff from across the UK is a horror story.
"It will be a specially bitter pill for staff to swallow as RBS has decided to move some of the jobs abroad to the Far East, India and America."