By Agence France-Presse
The newspaper said the inquiry was launched about two weeks ago following a complaint made to the MLB commissioner's office by Brian Cashman, general manager of the Yankees.
The Red Sox currently lead their division, the American League East, with the Yankees 2.5 games behind.
The Times, citing people briefed on the matter, said the Red Sox allegedly stole hand signals from opposing catchers in games against the Yankees and other teams.
In baseball, the catcher uses the fingers on his non-throwing hand to signal to a pitcher what type of pitch to throw.
A batter who knows what type of pitch is coming conceivably has an advantage.
Stealing signs is a venerable practice in professional baseball but the Red Sox are being accused of taking it to another level by introducing electronics.
According to the Times, the Red Sox admitted that a trainer in their dugout wearing an Apple Watch had received signals from video replay personnel and then passed that information on to some players.
It said the team's manager, John Farrell, was not aware of the sign-stealing scheme.
The newspaper said the Red Sox had filed a counter-complaint against the Yankees on Tuesday, accusing the New York team of using a television camera to steal signs.
The Times said it was unclear what action, if any, MLB commissioner Rob Manfred will take in response to the complaints.