CHICAGO—A Pakistan-born doctor with alleged ties to the 2008 Mumbai terrorist attack will be tried in a Chicago federal court starting Monday in a case that threatens to test already strained relations between the U.S. and Pakistan.

Prosecutors say Tahawwur Hussain Rana helped his high-school friend, Pakistani-American David Coleman Headley, conduct surveillance ahead of a three-day attack that killed more than 160 people, including six Americans. Mr. Rana also is accused of helping Mr. Headley in a bomb plot against a Danish...