The 30-year-old forward had 17 points (seven goals, 10 assists) in 64 games for the Oilers and Chicago Blackhawks this season, including four points (one goal, three assists) in 17 games with Edmonton after he was acquired in a trade on March 4.
Dickinson, who could have become an unrestricted free agent July 1, had three points (two goals, one assist) in four Stanley Cup Playoff games. The Oilers lost to the Anaheim Ducks in the Western Conference First Round in six games.
"It was fantastic, I loved playing here," Dickinson said after Edmonton was eliminated. "The crowd was amazing, the organization treats you fantastic, the city feels like home. I was saying to my wife, it kind of feels like home; driving around home playing hockey, everything feels so natural.
“I love it here and it's something I'd be willing to talk to them about and get something moving."
Selected by the Dallas Stars in the first round (No. 29) of the 2013 NHL Draft, Dickinson has 172 points (75 goals, 97 assists) in 566 regular-season games for the Stars, Vancouver Canucks, Blackhawks and Oilers and 12 points (seven goals, five assists) in 44 playoff games.
Edmonton is the only team in the NHL without a coach; Kris Knoblauch was fired on May 14, two weeks after it was eliminated by Anaheim.
NHL.com staff writer Derek Van Diest contributed to this report
