Gagner, 24, recorded 10 goals and 27 assists for 37 points in 67 games with the Oilers this past season.
In 481 career NHL regular-season games, all with the Oilers, Gagner has accumulated 101 goals and 194 assists for 295 points, along with 244 penalty minutes.
Gagner, a native of London, Ontario, was originally drafted sixth-overall by the Oilers in the 2007 NHL Entry Draft.
Crombeen, 28, recorded three goals and seven assists for 10 points, along with 79 penalty minutes, in 55 games with the Lightning this past season.
In 387 career NHL regular-season games with the Dallas Stars, St. Louis Blues and Lightning, Crombeen has accumulated 31 goals and 43 assists for 74 points, along with 771 penalty minutes.
The Denver native was originally drafted by the Stars in the second-round, 54th overall, of the 2003 NHL Entry Draft.