Jose Abreu has enjoyed his first look at Fenway Park, the historic venue that very nearly was his home.

While the first-time All-Star said Tuesday he left the free agency process that brought him to the White Sox to his agent, Barry Praver, multiple sources said the Boston Red Sox were a very serious player for the first baseman’s services.

Ultimately, Abreu signed a six-year, $68-million deal with the White Sox, who have begun to emerge from the depths of a 99-loss season in part because of how the slugger has brought vitality to an offense that failed to score 600 runs in 2013.

White Sox general manager Rick Hahn told the Boston Herald on Monday he knew of the Red Sox interest in Abreu, an attraction that was in part kept out of the spotlight because of Boston’s postseason run.

Given how Abreu has energized the franchise, Hahn doesn’t know how the White Sox offseason would have gone had the Cuban free agent wound up in Boston, which reportedly offered only $5 million less.

“It’s tough to know exactly what would have followed had we not been able to convert on that deal,” Hahn said on Tuesday. “But it certainly addressed a very important hole for us and plugged a need for the next several years and allowed us to sort of move on to some other (moves). Whether we would have been able to plug that hole through other moves, hard to say. But certainly its pretty safe to say it wouldn’t have been as strongly as we were able to by getting the deal done.”

An All-Star in his first season, Abreu’s arrival was significant for several reasons. First and foremost, the White Sox added a middle-of-the-order bat at the start of his physical prime, a player Paul Konerko has described as a rookie without being a rookie.

Given this is Abreu’s first go-round through the league, the White Sox feel as if they’ve tapped into a gold mine.