Callahan, 28, is making $4.825 million this season, though his cap hit is $4.275 million, and will become an unrestricted free agent July 1 if he goes unsigned.
The native of Rochester, New York, recently turned down a five-year, $30 million contract offer pitched by the Rangers, and is reportedly looking for a seven-year contract worth between $6.5 million and $7 million, annually.
If the Rangers are unable to sign Callahan, they are expected to trade him before the March 5 trade deadline.
The Olympic roster freeze, which many teams are looking at as a soft trade deadline before the official one, kicks in Friday at 3pm ET.
The St. Louis Blues, Vancouver Canucks, Detroit Red Wings and Phoenix Coyotes have all expressed an interest in Callahan.
In 40 games with the Rangers so far this season, Callahan has collected 11 goals and 13 assists for 24 points, along with 14 penalty minutes and a plus-2 rating.