It appears that was far from just lip service by Epstein, with SI.com reporting Wednesday that the sides have "growing optimism" that a new pact can be struck within the next few weeks.
Citing sources "close to the negotiations," the Web site reported that both sides expect a deal to come to fruition.
The Red Sox have a club policy of not discussing any ongoing contract negotiations.
Epstein and Beckett have both said repeatedly that they'd like to keep negotiations private until a conclusion is reached one way or another.
Lackey's five-year, $82.5 million deal is probably a rough parameter of Beckett's market value.
Beckett's current contract expires at the end of this season. Typically, the Red Sox spend time in Spring Training negotiating with players who are entering their "walk year."
The Red Sox respect Beckett not just for what he has done on the mound in his four seasons with the club, but for his leadership on the pitching staff.
Jon Lester, in particular, has emulated Beckett's work habits the last few seasons.
"I mean, I think it would definitely help out this organization, not only on the field, but off, for however many years they sign him," Lester said earlier this spring. "He's helped, and I can only speak for me, but he's helped me out a lot as far as preparing and just being professional. And it's not necessarily because he sits down and says, 'This is what you need to do.' He just does it."
In 122 starts with the Red Sox, Beckett is 65-34 with a 4.05 ERA.
He helped fuel the club to a World Series championship in 2007 by producing one of the best postseason runs (4-0, 1.20 ERA) of all-time. Beckett was similarly brilliant in leading the Marlins past the Yankees in the '03 World Series.
Ian Browne is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.