Briere told the Philadelphia Daily News he is not interest in a trade, as he owns a no-movement clause, while the Senators are not looking to deal Alfredsson.
However, if the Flyers continue to slide down the standings and are out of the playoff picture by the April 3 trade deadline, they might reconsider their options and could try to convince Briere to accept a deal.
The St. Louis Blues have also inquired about Briere, along with a few other clubs.
Chiarelli -- who also tried to re-acquire right wing Michael Ryder from the Dallas Stars, only to see him get moved to the Montreal Canadiens -- has previously knocked on the door of Calgary GM Jay Feaster, asking about Flames captain Jarome Iginla.
The Flames are not looking to deal Iginla at this point, and would require his blessing on any move.
However, the Flames might be in a position to consider several major moves this season if they cannot climb into the playoffs by the end of March.
Several reports have suggested that Iginla would consider a move if the Flames are out of the race by April. The 35-year-old is in the final year of his contract and might look to join a Cup contender.