Brown becomes a unrestricted free agent after the 2022-23 season.
The Toronto native had 10 goals and 29 assists in 64 games with the Senators last season.
The Oilers will be the third Canadian team Brown will play for after he started his NHL career with the Toronto Maple Leafs.
A former linemate of Connor McDavid in the OHL with the Erie Otters, Brown was selected in the sixth round (156th overall) of the 2012 NHL Draft by Toronto.
The Leafs traded Brown to the Senators in a multi-player deal in July 2019.
Brown had 90 goals and 126 assists in 444 career NHL games.