Last year, Cigna Corp. and the New York hospital system Northwell Health discussed developing an insurance plan that would offer low-cost coverage by excluding some other health-care providers, according to people with knowledge of the matter. It never happened.

The problem was a separate contract between Cigna and NewYork-Presbyterian, the powerful hospital operator that is a Northwell rival. Cigna couldn’t find a way to work around restrictive language that blocked it from selling any plans that didn’t include NewYork-Presbyterian,...