The bank said it hoped to work with staff to manage the cuts
Moves by RBS to cut 9,000 backroom staff worldwide have been branded "truly devastating".
The troubled bank said the cuts were prompted by a downturn in business and the need to create a leaner structure.
The Unite union said the prospect of losing up to a further 4,500 in the UK was "appalling".
The SNP said the move was a great concern for workers in Edinburgh, where RBS is based, and the Tories urged measures to help affected staff.
The bank, which is now 70% owned by the UK Government, is trying to save £2.5bn over the next three years. As a majority shareholder, the government wants to see the institution back on an even keel within five years.
RBS claimed the actual number of jobs lost was likely to be "significantly lower" due to natural turnover, fewer agency staff and voluntary redundancies.
Rob MacGregor, Unite national officer, said: "The news that 4,500 RBS staff in the UK are to lose their jobs is truly devastating.
"Unite is appalled that thousands of people, who form the backbone of the RBS operations, are to be made redundant. These employees are totally blameless for the current position which RBS is in, yet they are paying for the mistakes at the top of the bank."
Mr MacGregor called on the government to urgently put in place a programme of action to protect jobs and said the union would argue for creative solutions to cope during the downturn.
The bank has already announced plans to cut 2,700 jobs this year and had warned at its annual meeting last week that this was "not the end of the story".
Scottish Government Finance Secretary John Swinney said: "It is extremely disappointing news that RBS is consulting on a new business plan which could involve job losses in Scotland."
Mr Swinney said he had been in contact with RBS who assured him that compulsory redundancies would be a last resort and that the impact would be lessened by redeployment and natural turnover.
"I made clear the Scottish Government would work extremely closely with the company to help those affected and minimise the impact in Scotland as far as possible," said Mr Swinney.
Shirley-Ann Somerville, SNP MSP for Lothians, whose area includes the RBS HQ on the outskirts of Edinburgh, said: "Today's announcement will be devastating for those working at RBS and will cause great concern across the city as we wait to find out where the job losses will take place.
"After RBS initially said only 2,500 jobs would go - these figures are a dramatic increase."
Scottish Conservative MSP for Edinburgh Pentlands David McLetchie said: "Obviously I am very concerned about the potential loss of jobs in Edinburgh, where I suspect no-one will be leaving on the same redundancy terms as the government approved for Sir Fred Goodwin.
"We must hope that over the two-year period, scaling down can largely be achieved through retirement, voluntary redundancies or staff moving on."
Stephen Hester, chief executive of RBS, said: "We want to be as open and transparent as possible and are announcing these plans at the earliest possible opportunity so that our employees can prepare for the future."
Liberal Democrat Treasury spokesman Vince Cable said: "Ordinary workers are now paying with their jobs for the unsustainable and irresponsible practices of RBS's previous management."
Scottish Secretary Jim Murphy said: "Companies are having to cut costs at the moment but this is disappointing news nonetheless. It will cause concern for many families and individuals."