Jeffrey Gundlach is locked in a protracted stare-down with an unlikely foe: Morningstar Inc.

Gundlach, who formed his investing firm DoubleLine Capital in 2009 and built it into one of the most successful fixed-income fund companies, is sparring with the research firm over what DoubleLine says have been a series of false and misleading statements about its flagship fund, according to people familiar with the matter.