The move will affect 280 employees in Leeds, with around 115 roles relocating to Manchester, Liverpool and Sunderland.
Unite said workers will be "devastated" by the news.
A Barclays spokesman said: "Barclays currently employs over 12,000 colleagues across the north of England over a number of sites.
"Due to changes in our business, we have informed 280 colleagues based at Millshaw Court in Leeds that their roles are at risk, with around 115 of those roles relocating to Manchester, Liverpool and Sunderland.
"We don't take these decisions lightly and will do everything we can to support these colleagues, working closely with Unite the union.
"In addition, the landlord at Millshaw Court has put the site up for sale with planning permission for residential development.
"We have secured a three-year extension to the lease until November 2021, and we will be keeping our remaining employees regularly informed as we explore alternative sites."
Unite regional officer Christian Ratcliffe said the decision was "deeply irresponsible", adding that the union was opposed to compulsory redundancies.
"Unite has made it clear to the bank that these plans will significantly damage both the business and the customer experience.
"The number of employees at this key Barclays strategic site was 1,500 in 2014, and the announcement today will leave less than 600 permanent employees in Millshaw. Employees will be devastated.
"Unite is also calling on Barclays to urgently identify an alternative local site following the news that the lease on the current building will end in 2021."