Burke, doubling as general manager of the Flames, told Sportsnet 960 The Fan on Tuesday that the club has offered pending unrestricted free agent forward Mike Cammalleri a new contract in advance of the NHL Trade Deadline.
"We've offered him an extension," Burke told the radio station. "We have not gotten a deal done. We will continue to talk and a trade is a possibility, no question about that, but re-signing him is a possibility too. I would not predict how that will come out. We'll see.
"We have not set a drop-dead date. We have not set that date. We have had discussions with Mike and with his agent, both in person. The last meeting I had was with Mike and his agent. Mike has been great for us. He's been a disciple of what we're trying to do. He's been good with the young kids. He's worked his tail off."
Cammalleri, who serves as an alternate captain for the Flames, is in the final season of a five-year, $30 million contract and has a limited no-trade clause. The 31-year-old confirmed the parties have discussed an extension but stopped short at offering details of the talks.
"We've been talking," said Cammalleri, who has 13 goals and 21 points in 41 games for the Flames this season. "Brian has presented an offer that way. I guess we'll continue to talk here. I'm not ignorant to the fact that there's a deadline coming that might have implications. Brian's been great as far as being really open and honest and the lines of communication have been good. I'm sure that will continue.
"He has been really good in conversations behind closed doors with myself and my agent. I don't think it's fair to shed any light.
"I think that's more of a private conversation and we could talk over a coffee one day about that, but it's not something for in front of the cameras. [It's] probably something we'll keep in private conversation."
Burke has already been busy on the Flames' free-agent front. He re-signed center Matt Stajan to a four-year, $12.5 million extension Jan. 20 and followed it up Feb. 8 by locking up defenseman Kris Russell with a two-year, $4.52 million contract.
As a result, Burke has two fewer assets to peddle at the deadline.
Prior to last season's deadline, former GM Jay Feaster started Calgary's rebuild by trading forward Jarome Iginla to the Pittsburgh Penguins for Ben Hanowski, Kenny Agostino and a first-round pick in 2013 (used to select Morgan Klimchuk). Three days later, Feaster traded defenseman Jay Bouwmeester to the St. Louis Blues for Reto Berra, Mark Cundari and the Blues' first rounder (Emile Poirier).
Cammalleri, who is in his second go-round with the Flames, could be the next major piece shipped out in an effort to speed up the process. But he's given himself the option to stay.
"Plain and simple, I do like it here," Cammalleri said. "I like the guys a lot and I like the coaching staff. I have a lot of respect for them and what they're doing. As far as it goes from there, just getting to know Brian, great. Ownership and [Flames CEO] Ken [King] has always treated us fantastic. I've got nothing but fantastic things to say about Calgary, especially the people here. That's all I've got to say about that.
"There's a lot of other facets that go into whether or not I stay."
The NHL Trade Deadline is March 5 at 3 p.m. ET.