Tellqvist had been a third-stringer this season and appeared in only one game, a 2-1 loss to New Jersey on Nov. 16. J.S. Aubin, who has been dressing as the backup to Andrew Raycroft for most games, moved ahead of Tellqvist in the pecking order with strong performances down the stretch last spring.
"This will give Raycroft and Aubin each a full net in practice, and we like what we're getting back," said Leafs GM John Ferguson.
Ferguson is familiar with Nash from his days as assistant GM in St. Louis when Nash played for the Blues.
"He's a real agitating and energy-type player," said Ferguson. "He's got the ability to contribute offensively when given the chance.
"He's a good character player, and we also like the fact we've augmented our number of draft picks for next year. It was the right deal at the right time."
Nash, 31, had been playing for the San Antonio Rampage of the AHL. He had six goals and six assists in 19 games. He'll report to the AHL's Toronto Marlies.
The five-foot-11, 191-pound native of Edmonton has been hampered by injuries the last two years.
"Hopefully, he'll force our hand to take a look at him to see if he can help us at this level," said Ferguson.
Nash had 64 points (27-37) and 673 penalty minutes in 374 NHL games with St. Louis and Phoenix before being demoted to the AHL this autumn.
Tellqvist was 16-16-2 with the Leafs for his career with a 3.01 goals-against average and .895 save percentage. The 27-year-old Swede was happy to be getting a change of scenery.
"Michael has been a consummate professional," said Ferguson. "We wish him nothing but the best.
"He'll get an opportunity (in the Coyotes organization) and he's earned it. I know he's excited about it. He's a proven NHL backup, and maybe more."
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 different 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."
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.