The 27-year-old puck-moving rearguard can hit the open market as an unrestricted free agent on Sunday and is expected to command a deal with a cap hit of at least $4.5 million per year.
Carle could, however, have to wait until Sunday for the Flyers to complete a deal as the new salary cap will be in effect then.
If the Flyers can re-sign Carle, they will likely pair him up with newly-acquired Schenn.
"All I'll say is that we have a sincere interest - and the club does - to try to get it done," Carle's agent, Kurt Overhardt said.
Holmgren also added that if a deal can be reached with Carle, he likely won't be in the market for another defenseman, passing up on University of Wisconsin blueliner Justin Schultz, who is among the most coveted free-agent defensemen.
Meanwhile, Holmgren still believes there is a 50-50 chance that he will sign veteran forward Jaromir Jagr to a contract extension.
Although the GM has admitted that "losing James does create a hole, possibly," he added that he is not worried about the Flyers' offense.
"We have people internally who can do OK," he said. "And we will continue to look and do everything we can to improve the club."
The Flyers are expected to have a look at free agent Zach Parise of the New Jersey Devils and tradable Anaheim Ducks sniper Bobby Ryan and Columbus Blue Jackets captain Rick Nash, like several other teams around the league.
The team will also be shopping for a backup netminder to replace Sergei Bobrovsky, who was moved to the Blue Jackets last week. Holmgren is likely to inquire about former Flyers who can become free agents such as Martin Biron, Michael Leighton, and Antero Niittymaki.