The Dodgers trailed 7-0 against the Reds after three innings. Elly De La Cruz, Joey Votto and Matt McLain homered for the Reds in the rout, and Cincinnati left town in sole possession of first place in the National League Central.
In his postgame media availability, Dodgers manager Dave Roberts labeled Smith as day to day but added that he didn't expect it to be serious. Martinez, meanwhile, was slated for an MRI, Roberts said, before any decision on his timeline could be made.
Chris Taylor pinch hit for Martinez, who missed a pair of games last week because of a hamstring strain and then hit a home run in his return Tuesday.
Martinez, 35, is batting .260 with an .872 OPS in his first season with the Dodgers and has 25 home runs with 75 RBIs. A five-time All-Star in the American League, Martinez was named to the NL All-Star team for the first time this season. He is one of four Dodgers players with at least 20 home runs.
In 13 major league seasons with five different clubs, Martinez is a career .286 hitter with an .872 OPS and has 307 home runs with 974 RBIs in 1,493 games.
Smith, 28, is batting .279 with an .860 OPS, 13 home runs and 52 RBIs while earning his first NL All-Star nod. In five seasons with the Dodgers, Smith is a career. 265 hitter with an .857 OPS, 85 home runs and 282 RBIs in 437 games.
Despite the loss, the Dodgers are still in first place in the NL West, two games ahead of the San Francisco Giants.
Information from Reuters contributed to this report.