The 24-year-old goalkeeper has been linked with a potential move to Ibrox.
The Athletic claims that Rangers, alongside several English Championship clubs, are 'keen' on acquiring Rushworth.
However, the same source adds that Brighton have tabled a new deal for the shot-stopper, whose future remains 'clouded by uncertainty'.
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Since joining the Premier League side from Halifax Town in 2019, Rushworth has not made a senior appearance for the Seagulls.
Instead, he has spent most of his time on loan, amassing over 100 senior appearances at Walsall, Lincoln City, Swansea City, and Hull City.
He also has one England under-21 appearance to his name.
Despite having a contract at the AMEX until the summer of 2027, it is unclear where Rushworth will play next season.