A three-time 20-goal scorer, Vermette has another three years remaining on a contract that pays him $3.75 million annually.
"We are extremely pleased to acquire Antoine," Coyotes general manager Don Maloney said. "He is a smart, skilled, two-way center who will be a great addition to our lineup."
This season, the 29-year-old Quebec native has registered eight goals and 27 points in 60 games.
Selected by Ottawa in the second round (55th overall) of the 2000 NHL Entry Draft, Vermette has totaled 141 goals and 178 assists in 600 career games with the Senators and Blue Jackets.
Columbus obtained a 2012 second-round pick and a fifth-rounder in 2013 in the deal. The latter pick will become a fourth-rounder if Phoenix qualifies for the playoffs and advances to the second round.
McElhinney, 28, has compiled a 19-26-4 record with a 3.10 goals-against average and a .899 save percentage in 69 career NHL games -- 44 starts -- with Calgary, Anaheim, Ottawa and Phoenix. He saw action in two games with the Coyotes this season and is currently recovering from abdominal surgery.