Tellqvist, 27, appeared in only one game with the Leafs this season, a 2-1 loss to New Jersey on Nov. 16. He was 16-16-2 with the Leafs for his career with a 3.01 goals-against average and .895 save percentage. J.S. Aubin came in late last season and moved ahead of Tellqvist in the pecking order. Aubin has been the backup to Andrew Raycroft this season.
The Coyotes were looking to beef up their goaltending and see a good fit with Tellqvist.
"It's one of the most difficult positions in the National Hockey League to be a goaltender with the Toronto Maple Leafs," Coyotes GM Mike Barnett told The Canadian Press. "He's going to come to this situation without the enormous scrutiny and pressure that comes with playing that role in Toronto.
"Hopefully he'll thrive under a difference scenario and play more regularly."
Veteran Curtis Joseph remains the No. 1 goalie but Tellqvist "has a great opportunity here," added Barnett. "The opportunity is there for the taking."
Nash, 31, has spent the season with San Antonio of the AHL where he had six goals and six assists in 19 games. Known throughout his career as an agitator, the 5-11, 191-pound Nash has 64 points (27-37) and 673 penalty minutes in 374 career NHL games with St. Louis and Phoenix.
Also Tuesday, Maple Leafs defenceman Jay Harrison cleared NHL waivers and was assigned to the AHL's Toronto Marlies.
The 24-year-old appeared in five games with the Leafs this season, posting a minus-5 rating and six penalty minutes. He has one assist and eight penalty minutes in 13 career NHL games.