Its parent company, Aviva, said that about half of the job reductions would come through compulsory redundancies.
Up to 1,000 of the cut jobs will go to India and 500 IT jobs would be outsourced. David Fleming, national officier for the workers union Amicus, said: "This is absolutely brutal and compulsory job cuts and offshoring will not be accepted by us or our members.
Norwich Union employs about 36,000 people in the UK and the restructuring plans will trim about 11% of its total staff.
"This is a betrayal of Norwich Union's long serving workforce who have woken up to news in the media this morning that their jobs are going, rather than hear it from their employer.
“The fact that they are offshoring half of the work and sub-contracting some of that will have service implications for customers as well as staff.
"They are treating their staff with contempt and clearly have more regard for their shareholder profits then their UK workforce."
Today's job cuts include:
2000 redundancies compulsory
150 Stevenage, Bristol
100 Worthing, Belfast
"We only have three costs - people, technology and buildings," said Patrick Snowball, Norwich Union's executive chairman. "As you use more technology, you need fewer people and buildings."
Aviva and unions will now start a consultation period, with Amicus saying it would ‘consult with our members to see what they want to do’.