Davidson signed a one-year contract with the Blackhawks on Thursday that will net him $650,000.
Davidson will attend Chicago's training camp on a professional tryout offer, Scott Powers of The Athletic reports.
Davidson turned down a professional tryout offer from the Oilers, Reid Wilkins of 630 CHED reports.
Davidson had a goal, assist, and was a minus-8 in 15 games for the Islanders this past season.
Davidson (undisclosed) will be active for Saturday's home game against the Blackhawks.
Coach Doug Weight didn't mention Davidson (undisclosed) among the lineup changes Thursday, suggesting he will sit out the evening's affair against the Penguins.
Davidson will not play in Sunday's game against the Hurricanes due to a minor injury, Andrew Gross of Newsday reports.
Davidson will make his debut for the Islanders on Wednesday versus the Canadiens.
The Islanders acquired Davidson from the Oilers in exchange for a 2019 third-round draft pick Saturday.
Davidson scored his first two goals of the season and dished out five hits in Thursday's shootout win over Calgary.