Cano, a second baseman, has a quadriceps strain. McNeil has a left hamstring strain.
Cano has struggled in his first season with the Mets and was criticized for not hustling in two games when the Mets were swept by the Miami Marlins last weekend. He is hitting .241 with three home runs and 13 RBIs.
He suffered the injury hustling to first base on a grounder in the third inning of Wednesday night's 6-1 victory over the Washington Nationals. He had an MRI during the game, which showed a low-grade strain.
McNeil, who has started games in the outfield, at third base and at second base this season, is third in the National League with a .333 average. He also has two home runs and 15 RBIs.
McNeil has a tight left hamstring, which he thinks occurred while trying to steal second base in the ninth inning of a 6-5 win Tuesday. New York hasn't sent him for any image testing, and manager Mickey Callaway and McNeil both described the IL move as precautionary.
With its top two second base options out, the Mets expect to shuffle through options at the keystone. Adeiny Hechavarria started there Thursday, but Callaway described it as a "day-by-day'' decision.
The Mets now have four starters sidelined, as outfielders Michael Conforto (concussion) and Brandon Nimmo (neck) also are on the injured list.
Conforto is eligible to return from the seven-day concussion list Friday, but Callaway said he is unsure if the outfielder would be cleared. Conforto has been participating in baseball activities and was set to do so again Thursday, but he will need to be evaluated Friday before a decision can be made.
Pete Alonso, Wilson Ramos, Amed Rosario and Juan Lagares are the only position players from the Opening Day lineup who aren't injured.
Despite the health woes, New York has won three straight over the Nationals, getting key contributions from backups such as J.D. Davis, Carlos Gomez and Hechavarria. Rajai Davis was called up from Triple-A late Wednesday, arriving via Uber from Allentown, Pennsylvania, in the third inning, and hit a key three-run homer as a pinch hitter in the eighth.
"The guys aren't giving up,'' said Callaway, who got a front office endorsement Monday amid speculation about his job security. "Our depth is really stepping up.''
Infielder Luis Guillorme and left-handed reliever Ryan O'Rourke were recalled from Triple-A Syracuse to take the injured players' places on the roster.
In other moves, the Mets claimed outfielder Aaron Altherr off waivers from the San Francisco Giants and designated Tim Peterson for assignment
The Associated Press contributed to this report.
Cano leaves game with quad tightness
Robinson Cano injures himself while running out a grounder in the third inning and exits the game.