The trade is contingent upon Langkow passing a physical examination.
Langkow was limited to four games last season due to an upper body injury. He registered 14 goals and 23 assists during the 2009-10 season with the Flames.
The 5-foot-11, 185-pound Langkow has recorded 259 goals and 383 assists in 1,017 career games with Calgary, Phoenix, Philadelphia and Tampa Bay.
Langkow also has posted 14 goals and 23 assists in 59 career playoff games.
"We are very pleased to welcome Daymond back to the Coyotes organization," Coyotes general manager Don Maloney. "He is an intelligent, well respected, veteran center. His competitive, two-way game is a great fit for us and we are excited to have him back in Phoenix."
Langkow previously played for the Coyotes for three seasons from 2001-2004.
Stempniak had 33 goals and 23 assists in two seasons with the Coyotes (2009-11).
Originally drafted by St. Louis in the fifth round of the 2003 draft, Stempniak has 115 goals and 132 assists in 456 career games with the Blues, Maple Leafs and Coyotes.
"It is never easy moving a player who battles and competes such as Daymond; however, because of our depth and options at center ice, this trade presents us with an opportunity to positively impact the organization in a number of ways," Flames general manager Jay Feaster said. "It makes us younger and provides an opportunity to evaluate a player who may factor into our plans for the future. It provides Brent [Sutter] with options as Stempniak can play either wing and can be used on both the power play and the penalty kill, and it creates healthy competition among the forward group for ice time."