The New York Times, citing unnamed attorneys with knowledge of the test results, said Boston star Ortiz and Ramirez, who plays for the Los Angeles Dodgers, were among 100 Major League Baseball players to test positive in 2003.
Doping by Ramirez and Ortiz would cast a stain upon the historic 2004 World Series triumph by the Red Sox, which ended the team’s 86-year title drought.
The 2003 tests were meant to remain anonymous, serving only as an indicator of the need for a name-linked programme that began the next season.
But test results and information linking samples to players were seized by federal agents investigating the BALCO steroid scandal event as the US government and players’ union fight over the disposition of the materials.
In the meantime, Ortiz and Ramirez become the latest names that have been linked in reports to positive tests even though a court order has sealed the test information.
Others who have been reported as linked to positive tests from the 2003 testing include all-time US home run king Barry Bonds, New York Yankees star Alex Rodriguez, Dominican slugger Sammy Sosa, Jason Grimsley and David Segui.
Exactly which drugs the players tested positive for using was not revealed.
Ramirez completed a 50-game suspension earlier this month for violating league drug policy. Ortiz had not been previously linked to doping and the Dominican star has been one of the most popular Red Sox players.
Ramirez was named the 2004 World Series Most Valuable Player while Ortiz was named MVP of the 2004 American League Championship Series, in which the Red Sox rallied from a 3-0 deficit to win the best-of-seven play-off series over the arch-rival New York Yankees.
Ortiz and Ramirez also played key roles for the Red Sox in a 2007 World Series sweep of Colorado Rockies.