A source also said Dempster turned down a two-year, $26 million offer from the Kansas City Royals.
Boston's offer for Dempster would have paid him $12.5 million per season in 2013 and '14.
Dempster was a mainstay of the Cubs' rotation for nine seasons before being dealt to the Texas Rangers at the trade deadline last July. He struggled initially with the Rangers, then reeled off five consecutive wins before tailing off at the end.
Dempster, 35, had four straight seasons of at least 200 innings for the Cubs, with whom he had a 2.25 ERA in 16 starts last season before being traded. His stint with the Rangers was his first in the American League after spending nearly 15 seasons in the National League. He broke in with the Florida Marlins and also pitched two seasons for the Cincinnati Reds.
Two stints on the disabled list this past season cost Dempster a fifth straight season of 200 innings. He missed time early in the season with a strained right quadriceps, and in mid-June he was sidelined for 19 games with a strained lat muscle.
Dempster was charged with eight earned runs in two of his first three starts with Texas, but then went 5-0 with a 1.91 ERA in his next five starts, striking out 36 in 33 innings.
He wound up with 70 strikeouts in 69 innings for the Rangers, and his 9.1 strikeouts per nine innings with Texas were the most of his big league career.
Earlier in the week, the Red Sox made an offer that was described by a source as "competitive" for right-hander Dan Haren, who ended up signing a one-year, $13 million deal with the Washington Nationals. Another source said he believed Haren decided he had a better chance to re-establish his market value -- and pitch in the postseason -- for the Nats than he did with the Red Sox.
Comcast SportsNet Philadelphia confirmed the Red Sox offered center fielder Jacoby Ellsbury for Philadelphia Phillies pitcher Cliff Lee but were turned down. A source earlier this week said there was "not a prayer" of that deal happening, and subsequently, the Phillies acquired center fielder Ben Revere from the Minnesota Twins.
The Sox are not believed to be in the bidding for right-handers Zack Greinke and Anibal Sanchez, both of whom are expected to receive multiyear contracts for more years than the Red Sox are willing to offer. Another top free agent, Kyle Lohse, is represented by Scott Boras and is expected to be seeking more than three years, though if a three-year deal is sufficient to get him signed, the Red Sox would be expected to jump in.