PORTLAND, Ore.—Nike Inc. and Under Armour Inc. have long fought for shelf space at retailers. Lately, though, they have been quibbling over another coveted spot: a weed-strewn running track in Nike’s backyard.

The track and field at Duniway Park here has become an unlikely pawn in the companies’ game for control of the sportswear market. Baltimore-based Under Armour plans to move its global footwear operations into a building adjacent to the public park and has offered to fund a renovation at Duniway. In September, local officials...