The 27-year-old forward was in Arizona training camp on a professional tryout contract after scoring 13 points (five goals, eight assists) in 34 games with the Toronto Maple Leafs and Ottawa Senators last season. He scored four points (one goal, three assists) in six Stanley Cup Playoff games for Toronto.
"We are very pleased to sign Alex," Coyotes general manager Bill Armstrong said. "He is a versatile, skilled forward who works hard and has earned a spot on our roster. He will be a good addition to our team."
Galchenyuk played for the Coyotes in 2018-19, scoring 41 points (19 goals, 22 assists) in 72 games. He was sent to the Pittsburgh Penguins on June 29, 2019, as part of the Phil Kessel trade.
Selected with the No. 3 pick in the 2012 NHL Draft by the Montreal Canadiens, Galchenyuk has scored 333 points (140 goals, 193 assists) in 583 games with the Maple Leafs, Senators, Minnesota Wild, Pittsburgh Penguins, Coyotes and Canadiens, and 17 points (five goals, 12 assists) in 38 playoff games.