The 24-year-old forward could have become a restricted free agent Wednesday.
Gaudette scored 11 points (five goals, six assists) in 40 games this season, including four (one goal, three assists) in seven games for Chicago after being acquired in a trade with the Vancouver Canucks for forward Matthew Highmore on April 12.
Selected by the Canucks in the fifth round (No. 149) of the 2015 NHL Draft, Gaudette has scored 56 points (22 goals, 34 assists) in 160 regular-season games, and no points in 10 Stanley Cup Playoff games.
Gaudette won the Hobey Baker Memorial Award voted as top men's player in NCAA Division I ice hockey in 2017-18, when he scored 60 points (30 goals, 30 assists) in 38 games as a junior for Northeastern University.