Cole has 97 points (20 goals, 77 assists) in 405 NHL games during eight seasons with the St. Louis Blues, Penguins and Blue Jackets. He did not appear in a game with the Senators.
Cole played on the second defense pair with David Savard in Columbus. He was also regularly deployed on the top penalty-killing unit.
"My agent has been in contact with the management of the Blue Jackets a couple of times, back and forth," Cole said. "Nothing to comment on as of yet, but I'm sure things will pick up as we get closer to July 1."
He told the website he enjoyed playing and living in Columbus.
"My perception of Columbus … for the most part, we didn't spend much time there," he said. "Fly in, go to dinner somewhere near the hotel … get up the next day, play the game, hear the cannon fire a few times and leave. I didn't really know too much about it.
"But getting to know the city, to experience it … it was really eye-opening. It's a great city, with great people. I didn't know much about it, but I left there very happy with it and very excited about the possibilities down the road."