Nike, the world's largest sportswear manufacturer, says it has 19 separate patents protecting its SHOX system.
Adidas, the second largest sportswear company, completed its takeover of Reebok in January.
The company paid $3.8bn (£2.2bn) for the American firm to better its competition with Nike and to improve its North American market share.
"Despite Nike's patent protection, Adidas has built shoes that use Nike's technology," Nike spokesman Vada Manager said on Thursday, but on Friday morning Adidas said they would be making a statement regarding the Nike action.
Nike, which in January lost former chief executive William Perez who left after a boardroom row, has made the legal complaint against Adidas in a court in Texas.
The Adidas trainers Nike claims infringe its patents all use Adidas' a3 cushioning system it is claimed. One example of such a shoe is the new Kevin Garnett signature shoe, named after a popular American basketball player.
