The Times said Sosa is one of 104 players who tested positive in baseball's anonymous 2003 survey, which has been the subject of a protracted court fight. The paper did not identify the drug.
It cited lawyers with knowledge of the 2003 results and reported they spoke on condition of anonymity because they did not want to discuss material under court seal.
In 2003, baseball did not have penalties for the first-time use of performance-enhancing drugs.
Sosa's agent, Adam Katz, said he had no comment. Commissioner's office spokesman Rich Levin also had no comment, saying the league didn't have a copy of the test results.
Michael Weiner, the union general counsel, also declined comment.
The union, while fighting to get the list back from the government, has mostly refused to discuss reports about the list because it does not want to confirm or deny who is on it.
Several of the game's biggest stars, including home-run king Barry Bonds, Roger Clemens, Mark McGwire and Jose Canseco, have been implicated in steroids use.
Sosa sat alongside Rafael Palmeiro, Canseco and McGwire at a March 2005 hearing before Congress and testified: "To be clear, I have never taken illegal performance-enhancing drugs."
"I have never injected myself or had anyone inject me with anything," he said. "I have not broken the laws of the United States or the laws of the Dominican Republic. I have been tested as recently as 2004, and I am clean."
That left open the possibility he used a substance legally in the Dominican Republic that would have been illegal to use in the United States without a prescription.
Rep. Henry Waxman, the co-chairman of the hearing, declined comment, spokeswoman Karen Lightfoot said.