The business, which is being reviewed as part of a statutory process, is currently held by Bright. The agency has been invited to repitch.
The Aviva-owned company has contacted ten agencies and the pitch is expected to be held in September.
Bright won the business in 2003. It was previously held by Draft. Bright handled customer communications for the company's consumer and corporate divisions, as well as relationship marketing.
Norwich Union Healthcare uses direct mail, press inserts, door-to-door work and online across its insurance products.
Last year, the insurer had a campaign that highlighted long waiting times for operations at two NHS hospitals banned by the Advertising Standards Authority. The ads were criticised for exaggerating the situation at the Royal Cornwall and Royal Shrewsbury Hospital Trusts.