The Times, citing lawyers familiar with the case, reported Sosa was one of 104 players who tested positive in a 2003 MLB survey. The paper did not identify the drug.
Sosa is sixth on MLB's career home run list with 609. He has not played in the majors since 2007.
Sosa's agent, Adam Katz, told The Associated Press he had no comment on the report. MLB commissioner's office spokesman Rich Levin also had no comment, saying MLB didn't have a copy of the test results.
Several of the league's biggest stars, including home-run king Barry Bonds, Roger Clemens and Mark McGwire, have been tainted by steroids scandals. Los Angeles Dodgers slugger Manny Ramirez is serving a 50-game suspension for violating MLB's doping policy; just a few months ago, New York Yankees star Alex Rodriguez admitted using steroids from 2001-03 with Texas.
Former pitcher Pedro Martinez played against Sosa for many years.
"This news would make me feel terrible if it is proven that Sammy tested positive," Martinez said in the Dominican Republic.
"This is a problem of all of baseball, not just Dominican baseball. But in reality, this is a problem of education that has to be attacked."
Sosa testified before the U.S. Congress in 2005 and denied any wrongdoing, saying, "To be clear, I have never taken illegal performance-enhancing drugs."
At that hearing, Sosa sat beside Rafael Palmeiro, Jose Canseco and McGwire, all three of whom have been blemished by doping.
Sosa told the hearing: "I have never injected myself or had anyone inject me with anything. 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.
Sosa, now 40, and McGwire engaged in a race in 1998 to break Roger Maris' MLB single-season record of 61 home runs, a chase that captivated the U.S. McGwire set the mark while Sosa, with a big smile and a trademark hop out of the batter's box, finished with 66.
Sosa followed up by hitting 63, 50, 64, 49 homers in his next four years. He hit 40 more in 2003, a season in which he was caught using a corked bat.
MLB prohibited the use of steroids without a valid prescription from 1991, but did not reach an agreement with the union until 2002. There were no penalties for a positive test in 2003 - those tests were conducted to determine if it was necessary to impose mandatory random drug testing across MLB in 2004.
As part of the drug agreement, the results of the anonymous testing of 1,198 players in 2003 were meant to be anonymous. Penalties for a first positive test did not start until 2005.