Darek Braunecker, Burnett's agent, confirmed that there have been discussions with the Red Sox during the Winter Meetings.
In fact, he described it as "significant dialogue" to The Boston Globe, adding that Monday's conversations with the Red Sox were "certainly more substantive than our last communication."
In later group settings with media members on Tuesday, Braunecker chose to get less specific in regards to Boston's interest in his client, but noted one priority of Burnett's that would certainly make the Red Sox a top fit.
"Winning is the most important ingredient," Braunecker said. "Geography is not the issue that I thought it was going to be coming into this, but the personnel within the club and, most importantly, winning [is]. Which club provides him the best opportunity?"
The Red Sox have been to the American League Championship Series four times in the last six years and have won the World Series twice in the last five years.
"He's 31 years old. He's been in the league now for nine years," Braunecker said. "He got a real taste of what winning and success was all about this year when he went on those couple of runs in Toronto this year. Now he understands how much fun winning is. [With the Marlins in 2003] didn't count, because he wasn't a part of that team. That's one of the primary criteria that he's looking for from his next employer."
The Red Sox aren't alone. The Yankees, Braves, Blue Jays, Orioles and others are also pursuing the righty. Atlanta is believed to be the most aggressive player for Burnett thus far. The Yankees will likely ratchet up their intensity if they can't land CC Sabathia.
If Burnett did come to the Red Sox, he would be reunited with principal owner John W. Henry, who served in the same capacity for part of the righty's time with the Marlins.
"Any time a player has a relationship with the owner, that can't be a bad thing," Braunecker said.
What are the Red Sox's chances of landing a pitcher who has electric stuff?
"It's impossible for me to handicap it right now," Braunecker said. "I really don't know. If I could tell you that, I would."
If Burnett is insisting on signing a deal of more than four years, it's likely he'll wind up somewhere else besides Boston. With the exception of Matsuzaka, who was 26 years old at the time, the Red Sox have never offered a free-agent pitcher more than four years during Theo Epstein's time as general manager.
During parts of Burnett's career, he has been marred by injuries. But he was a horse in 2008, making 35 starts for the Blue Jays. Burnett went 18-10 with a 4.07 ERA, notching 231 strikeouts.
As far as Braunecker is concerned, Burnett's durability is no longer an issue.
"[It] seems to me that everybody is beyond those issues and that he's learned his body," Braunecker said. "He's grown up, he's filled up. He's learned as a baseball player how to maintain his regimen throughout the course of a 162-game season to insure that he's able to take the ball every fifth day."
Epstein is already on record as saying he doesn't think he'll complete a deal in Vegas.
Will Burnett sign in Vegas?
"All I can say is that the process is progressing. At what point we get to the finish line, I don't know," Braunecker said. "I didn't come into this with the intent on getting something done [here in Vegas]. If the right deal presents itself by the right club, we'll act. At this point, we're not comfortable that we're at that stage."
Ian Browne is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.