Derby, where it employs about 12,000 workers, will bear the brunt of the cuts, with about 300 jobs set to go.
Another 140 jobs are set to be lost at its plant in Ansty, Warwickshire.
The proposed job losses are among 2,600 worldwide scheduled for the next 18 months, mostly from the firm's aerospace division.
Roll-Royce said restructuring of its turbine business would lead to the possible closure of a site in Derby and one at Ansty.
More than 150 jobs are also set to go at factories at Inchinnan, Renfrewshire; Barnoldswick, Lancashire and Hucknall, Nottinghamshire.
A spokesman for Rolls-Royce said: "As part of this restructuring process we're looking to make the best use of all our facilities.
"This would result in the closure of our turbine blade machining facility at Ansty and the precision machining facility in Derby during 2017.
"It is never an easy decision to make changes and propose reductions in our workforce, but it is a sign of the increasingly competitive market in which we operate.
"The measures we have outlined to our employees will contribute towards Rolls-Royce becoming a stronger company in the long term."
Unite union officer Tony Tinley said: "These cuts are a huge loss of skills to the UK economy and will result in Rolls-Royce outsourcing high-tech manufacturing jobs overseas to plug the skills gap
"There is a real danger that Rolls-Royce is making decisions in the short-term which it will later regret, and it needs to give a guarantee of no compulsory redundancies."
The union said the engineering firm needed to give assurances over its long-term future in the UK.
The company's UK staff are employed at four locations in the East Midlands, as well as 1,500 at five sites in the North West and 2,400 employees at six locations across Scotland.
Rolls-Royce said the job losses would cost it Â£120m over the next two years, but would bring "annual cost benefits" of Â£80m once implemented.
The company said voluntary redundancy would be offered wherever possible.
Hundreds of jobs are set to be axed at Rolls-Royce factories in the UK under plans for a worldwide cutback, it has been announced.