The state-owned bank said it was moving the jobs, which help to handle loans for small businesses, as part of an ongoing cost-cutting drive.
But the company, which owns RBS and NatWest, added that UK staff would still deal with customers and take the decisions on whether to grant loans.
The Unite union said UK workers and taxpayers would lose out from the move.
“By shipping these jobs to India, RBS will be getting that work done more cheaply at the cost of jobs and livelihoods here in the UK,” a spokesman said.
The bank’s small business customers will also be unsettled by the decision, according to Mike Cherry, chairman of the Federal of Small Businesses.
“Many small business customers with RBS will be extremely concerned at the idea of local expert staff being sent packing and their roles outsourced to call centres halfway round the world,” Mr Cherry said.
He added it was the “wrong way to rebuild trust” after branch closures and a mis-selling scandal that saw thousands of small business customers compensated by the bank last year.
The company, which is still 73% owned by the government after a £45bn bailout in 2008, said staff in Mumbai would take over back-office roles such as background checks.
But it added that UK-based staff would continue to do the work that involved customer contact.
Credit decisions will also be taken in the UK, according to the bank.
If you’ve just agreed a new mortgage or remortgaged, you’ll probably not have met the person who decided to give it to you (or not).
That’s not the case with small companies. They have a far closer relationship with their bank manager. The latter needs to know what kind of business you have and how you run it.
Some may have fretted that shifting more than 400 jobs relating to SME (small and medium-sized enterprises) lending to India might irrevocably alter than relationship.
And that’s why RBS is stressing that all credit decisions will continue to be taken here in Britain and no relationship managers with SMEs will be downsized.
RBS doesn’t want to damage its current position as the largest business bank in the UK.
A spokesman for the RBS group said: “As we become a simpler, smaller bank, we are making some changes to the way we serve our customers.
“Unfortunately, these changes will result in the net reduction of 443 roles in the UK.”
The company said it would support staff affected by the “disappointing news”, including by moving them into new roles where possible.
It comes just weeks after RBS said it would cut 250 IT jobs in the UK and move dozens of the roles to India.