It’s starting to look like an old-time coach swap is going to happen, and who would have guessed that three years ago?

With the Rangers getting ready to introduce Alain Vigneault as their new head coach today at 11 a.m. at Radio City Music Hall, the Canucks are closing in on John Tortorella as their new head man, now a finalist for the job with Jacques Martin and John Stevens. The Canucks fired Vigneault after being swept in the first round of this year’s playoffs by the Sharks, and the Blueshirts dismissed Tortorella after they lost to the Bruins in a five-game, second-round series.

There was also no comment yesterday from Mark Messier, who interviewed for the Rangers’ job but was passed over for Vigneault. General manager Glen Sather told The Post’s Larry Brooks on Wednesday that Messier was “very disappointed” to hear the news, and he had yet to respond in figuring out a possible future role with the team. Messier, the Rangers’ icon of the 1994 Stanley Cup-winning team, had been a special assistant to the general manager since 2009.

In February 2009, Tortorella began his 4 1/2 -year tenure behind the Rangers’ bench, and his defense-first, disciplinarian approach eventually turned sour. Following the organizational exit meetings in late May, Sather surprised Tortorella with his decision for dismissal, and it had a lot do with what was perceived as a style of play that had passed him by.

“If you look at the teams that are going to be contending for the Stanley Cup, there are specific ingredients that every team has,” Sather said on May 29, referencing the best teams’ penchant for puck possession and their ability to score goals. “I’d like to take the best of what those teams have and integrate them into our team.”

Tortorella’s teams did a lot of dumping the puck into the zone and chasing it down, starting their offense behind the other team’s net and scoring near the bottom of the league. Yet it was the style that got them to within two wins of the Stanley Cup final last season, when they lost to the Devils in overtime in Game 6 of the Eastern Conference finals.