The goalkeeper has been recently linked with a move to Ibrox.
New head coach Russell Martin has been active in the transfer market this summer, and securing a new stopper could be next up on his agenda.
While The Athletic claims that Rangers are 'keen' on Rushworth, along with clubs in the English Championship, it is said that Brighton have offered him a new deal.
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His future is 'clouded by uncertainty', with the 24-year-old having not made a senior appearance for the Premier League club since joining them from Halifax Town in 2019.
Rushworth has spent three and a half of the past four seasons on loan at Walsall, Lincoln City, Swansea City and Hull City, racking up over 100 senior appearances. He has also played once for England's under-21s.
The keeper's current deal at the AMEX runs until the summer of 2027, yet it remains to be seen where he'll be plying his trade next season.
Rangers have already signed six players this summer. Nasser Djiga became the latest arrival on Wednesday, following on from Lyall Cameron, Max Aarons, Joe Rothwell, Thelo Aasgaard, and Emmanuel Fernandez.