Burke has already reached out to one team to seek permission to speak with one of that team's personnel.
It's believed the Dallas Stars have granted the Flames permission to speak with former GM Joe Nieuwendyk about their new vacancy.
The Flames are expected to contact several other clubs later this week, as their search for a new GM officially begins.
Feaster, who was in the final year of his contract, joined the Flames in July, 2010 as assistant GM, and was later named acting GM on Dec. 28, 2010 before taking over as the club's full-time GM on May 16, 2011.
Feaster won a Stanley Cup Championship as General Manager of the Tampa Bay Lightning in 2004 and a Calder Cup Championship as the President of the Hershey Bears (AHL) in 1997.
The native of Williamstown, PA was originally named GM of the Lightning on Feb. 10, 2002. In addition to a Stanley Cup championship, the Lightning won back-to-back Southeast Division titles in 2002-03 and 2003-04.
Feaster was named the 2004 Sporting News NHL Executive of the Year based on a vote of other NHL GMs and hockey executives.
Weisbrod, a native of Syosset, NY, was in his second season as the Flames Assistant GM in charge of player personnel.
Prior to joining the Flames, Weisbrod was part of the Boston Bruins Stanley Cup championship team in 2011 where he was the Director of pro and collegiate scouting since joining the organization in Sep. 2006. Prior to that, he spent the 2005-06 season as a pro scout for the Dallas Stars out of the New England region.