Jaguars owner Shad Khan would be willing to make Ramsey the league's highest-paid cornerback as soon as possible, according to NFL Network's Ian Rapoport.
It's currently unknown whether the star defender would be willing to remain with Jacksonville if he received a lucrative contract offer. He previously said he would ask for an "ungodly" amount of money to re-sign with the club.
The 24-year-old's trade request came after a "disrespectful" conversation with decision-makers Tom Coughlin and David Caldwell.
Ramsey suited up for Thursday Night Football against the Tennessee Titans despite the fact that he reportedly doesn't expect to be with the Jaguars by Week 4.
The two-time Pro Bowler isn't short of suitors on the trade block. Jacksonville reportedly set its asking price for the cornerback at two first-round picks.