The lawsuit, filed Thursday in a federal court in Manhattan, accuses Ortiz and his sister of naming their Santo Domingo club Forty-Forty in the hopes of trading in on the rapper's name.
The suit seeks more than $5 million in damages and for Ortiz to forfeit any use of the name.
Jay-Z, a Yankees fan who performed at the team's World Series victory parade last year, opened his original 40/40 club in Manhattan with business partner Juan Perez in 2003.
He promoted the sports bar and lounge by dropping references to it in rap hits like "Dirt Off Your Shoulder." The pair subsequently opened up 40/40 clubs in Las Vegas and Atlantic City and have licensed the name to operators in Tokyo and Macau.
"David Ortiz is fully aware of plaintiff's Manhattan 40/40 club, since he had been a patron there on several occasions before he opened his Infringing Forty/Forty club," the lawsuit said
Court filings included photographs of Ortiz at the club.
SFX Baseball, the agency that represents Ortiz, did not respond to phone and e-mail messages.
The 40/40 club's lawyers also asked the court to block the slugger from using the Forty Forty name on his club's Web site.
Jay-Z's 40/40 club is named after the rare feat among ballplayers of hitting 40 homers and stealing 40 bases in one season.
Ortiz is a power hitter who has stolen 10 bases in 13 seasons, but he has hit more than 40 homers three times.