Multiple league sources have told TFP the Rangers inquired about St. Louis' availability over the last few seasons, but were told he wasn't available.
St. Louis, who lives in Connecticut in the off-season, has one-year left on his contract beyond this season. He requested a trade after being originally left on the Canadian Olympic men's roster.
The 38-year-old met separately with Lightning owner Jeff Vinik and Bolts GM Steve Yzerman after not initially making Team Canada. He was later named to the team as a replacement for the injured Steven Stamkos.
The Rangers first offered right wing Ryan Callahan for St. Louis, but Yzerman is not interested in moving his star forward for a rental player.
Talks between the Rangers and Bolts haven't progressed, though, according to the New York Post.
The Post claims Rangers GM Glen Sather is not willing to add young pieces and draft picks with Callahan for St. Louis.
Negotiations will continue until the trade deadline arrives on March 5 at 3pm ET.
If St. Louis is not traded this season, the two sides will try to complete a deal in the summer.