Rodriguez, who sat out games against the Tampa Bay Rays on Saturday and Sunday, was due to undergo the surgery at the University of Miami.
"There is no doubt we're going to miss him but we need to find a way," Yankees manager Joe Girardi said in a statement.
"As I've said several times already this year, some people are going to have to step up. Up to this point, this team has. We certainly look forward to getting a healthy Alex back into our lineup, because we know what he adds to this team."
Rodriguez, 35, hurt his knee running the bases against the Chicago Cubs last month and underwent tests Friday because of lingering soreness.
The Yankees trail the Boston Red Sox by one game in the American League East and the loss of 'A-Rod' is a blow to their hopes of gaining top spot.
Rodriguez, second on the team with a .295 batting average this season, played in 14 games after the injury.
(Reporting by Simon Evans in Miami; Editing by Frank Pingue)