The paper cited two people involved with baseball's drug-testing program, whom it did not identify.
Baseball's joint drug agreement specifies the discipline for a first positive test for a banned stimulant is six additional unannounced drug tests over the year following the violation. A second stimulant violation would result in a 25-game suspension.
Rodriguez spokesman Lanny Davis denied the player tested positive, the paper said. Ron Berkowitz, another Rodriguez spokesman, said Monday he expected his client's representatives would comment later in the day.
MLB chief operating officer Rob Manfred declined comment.
Rodriguez was suspended for 211 games by Major League Baseball on Aug. 5 under baseball's drug agreement for his alleged "use and possession of numerous forms of prohibited performance-enhancing substances, including testosterone and human growth hormone, over the course of multiple years." He also was penalized under the labor contract for "a course of conduct intended to obstruct and frustrate the office of the commissioner's investigation."
The three-time AL MVP was allowed to keep playing until arbitrator Fredric Horowitz decides a grievance filed by the players' association to overturn the penalty. Horowitz has presided over eight days of hearings, which are scheduled to resume Nov. 18.
MLB and the union agreed in November 2005 to ban many stimulants. The paper said it wasn't clear whether a failed stimulant test was introduced by MLB as evidence in the grievance.
Sports Illustrated reported in February 2009 that Rodriguez tested positive for steroids during MLB's anonymous survey in 2003, and A-Rod said two days later he used banned substances while playing with the Texas Rangers from 2001-03.