Rothwell came through Manchester United’s youth ranks and, after loan spells with Blackpool and Barnsley, joined Oxford United.
He moved on to Blackburn Rovers and then, in 2022, to Bournemouth.
Rothwell follows fellow free agent midfielder Lyall Cameron, 22, and on-loan Bournemouth full-back Max Aarons, 25, in joining Rangers this summer.
Martin’s side open their season at home to Panathanaikos in the Champions League second qualifying round first leg on 22 July.
