Globe & Mail reporter Bruce Dowbiggin first reported the news via Twitter, and indicated that Burke could possibly take over at the Flames' CEO.
TFP has confirmed that the Flames have had talks with Burke, though it's unclear how far they have gotten.
Late Wednesday night, TSN Hockey Insider Bob McKenzie also confirmed via Twitter that "Burke is deep into discussions to join the Flames in an executive position but it is not, as of this moment, done."
Also Wednesday night, CTV Calgary has reported that Burke is definitely joining the Flames' organization, citing a "reliable source."
The Flames have been trying to add a new high-ranking person to their managerial staff for several months. In addition to their constant communication with Burke, the Flames met with Hall-of-Famer Brendan Shanahan, but he opted to remain the NHL's VP of Hockey and Business Development, and Director of Player Safety.