Jon Heyman of FanRag Sports passed along the update Tuesday.
Martinez is one of the top players available on the open market after a monster 2017 season with the Detroit Tigers and Arizona Diamondbacks. The right fielder posted a .303/.376/.690 triple-slash line with 45 home runs in just 119 games.
In September, the 30-year-old Miami native told Nick Piecoro of AZCentral Sports he planned to seek out offers from championship contenders.
"I think one of the most important things in my free agency is I want to be on a team that's relevant – a team that's good, a team that's in it," he said. "I don't want to be on a team just because (they offer the most money). I want to be on a team that's got a chance. That's what's fun. That's what you play the game for."
While Martinez downplayed the importance of money, Bob Nightengale of USA Today reported last month the initial asking price from Martinez's representatives was a seven-year, $210 million contract.
Along with the Red Sox, the San Francisco Giants could also make a push to sign the slugger after missing out on Japanese dual-threat sensation Shohei Ohtani and Miami Marlins outfielder Japanese megastar Giancarlo Stanton, who was traded to the New York Yankees, per John Shea and Henry Schulman of the San Francisco Chronicle.
Boston is an ideal fit on paper. The Red Sox won the AL East with a 93-69 record last season but could benefit from the addition of another power bat after ranking 27th in homers.
Martinez would slide in Mookie Betts, Andrew Benintendi and Xander Bogaerts to give the team one of baseball's most dangerous top halves of a lineup.