Lloyds said it was cutting 1,600 positions as part of a broader plan announced last year to cut 15,000 jobs and halve the group’s international presence.
RBS said it was cutting a net 300 jobs approximately, with the reduction of 464 positions offset by 150 new jobs elsewhere.
Unite said it had written to UKFI — the body set up by the government to manage the state’s bank holdings — asking for the government to intervene.
“How can there be any justification for the government not intervening as these much needed jobs are lost from our struggling economy,” Unite national officer David Fleming said.
Both Lloyds and RBS said they would try and keep compulsory redundancies to a minimum.
“We are working hard to rebuild RBS in order to repay taxpayers for their support and having to cut jobs is the most difficult part of this process,” RBS said.
“We strive at all times to be open and honest about the tough choices we are making and we remain committed to providing our customers with the same high level of service as we make changes to our Private Banking Direct structure.”
Lloyds said that of the 1,600 affected positions, it was transferring some 300 workers over to its suppliers, meaning they would still have a job.
RBS has cut 34,000 jobs since 2008 while Lloyds has cut around 30,000.