The 30-year-old midfielder won promotion with Leeds United last season and has one year left on his current deal.
If a permanent move is agreed, Rothwell would sign a two-year contract with the Light Blues.
Rothwell played under boss Russell Martin at Southampton, helping them win promotion to the Premier League.
He would join the squad in the middle of the park alongside Lyall Cameron, Connor Barron, Nico Raskin, Mohamed Diomande, Nedim Bajrami, and Bailey Rice.
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Rangers have already signed one player from Bournemouth during the summer transfer window, right-back Max Aarons on a loan deal.
Lyall Cameron was another summer signing, also competing for a place in central midfield.
Elsewhere, Rangers are close to appointing former Arsenal goalkeeping coach Sal Bibbo as part of Martin's staff.