Girardi, 29, is scheduled to become an unrestricted free agent July 1 and contract negotiations between his agents and the Rangers have reportedly hit a stalemate.
According to the New York Post, the Rangers have informed other teams that they are officially willing to trade Girardi, instead of risk losing him for nothing as a free agent this summer.
The Ducks have multiple young assets they appear willing to trade, though it's unclear if Murray has any interest in moving a player like Sami Vatanen, whom the Rangers are believed to have interest in, as part of a deal for Girardi.
Several other teams are interested in Girardi, including the Boston Bruins, Los Angeles Kings and Philadelphia Flyers.
In 59 games with the Rangers so far this season, Girardi has tallied four goals and 13 assists for 17 points, along with 14 penalty minutes.
If the Ducks are unable to acquire Girardi, they could turn to New York Islanders blueliner Andrew MacDonald, who is highly coveted and is expected to be moved by the deadline.