That’s according to the Daily Record.
The Light Blues have made four signings this month with Jermain Defoe joining from Bournemouth on an 18-month loan deal and Steven Davis re-joining the club on a permanent basis.
Jordan Jones and Glen Kamara have also been snapped up on pre-contract agreements to move to Ibrox in the summer.
But with Old Firm rivals Celtic also busy in the January market, Gerrard could now look to make another permanent acquisition this month to strengthen their Scottish Premiership title push.
The Daily Record say that Rangers are seeking to sign Bryson on a free move this month with the 32-year-old out of contract in the summer.
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Gerrard’s right-hand man McAllister has recommended a move for Scotland international Bryson and seemingly rates his compatriot highly.
And Liverpudlian coach Gerrard has been convinced to move in for Bryson, who has made 18 Championship appearances for Frank Lampard’s promotion-chasing Derby.
Bryson – who spent last season on loan at Cardiff City – has however dropped to the bench in three of the Rams’ last four league matches.
And he has yet to be offered a new contract at Pride Park with less than six months remaining on his existing deal.
The Daily Record add that Derby chairman Mel Morris is keen to cut down the club’s wage bill and Bryson – their longest-serving player – could be sacrificed as a result.
Rangers are thought to have opened negotiations with Derby and hope to seal a free transfer that would allow them to bring Bryson in this month rather than wait until the summer.
Rangers have a host of midfield options including new addition Davis, Ryan Jack, Scott Arfield, Ross McRorie, Jordan Rossiter, Graham Dorrans and loanee Lassana Coulibaly – with Kamara also an option in the engine room from the summer.
But Gerrard has been tempted to make an approach for Bryson, who has made 263 appearances for Derby since joining from Kilmarnock eight years ago and turned 32 in November.
Speaking to Derbyshire Live before the New Year, Bryson said when asked if he had been offered a new deal: “No, there hasn’t been anything. It’s up to the club. It’s in the club’s hands.
“I’ve got six months left of my deal. If they don’t come to me with anything, then it’s time to move on. As it stands, that’s something that could happen.”
Gerrard has declared he wants to bolster his squad for the immediate future and the long-term future of the Glasgow giants.
He revealed: “Jordan and Glen are signings for the big picture but I’ve got a responsibility to try and improve the team now and get results now but also think about trying to strengthen the squad for the years to come,” Gerrard said.
“They are two really good signings in terms of potential and very excited to come but we’ve asked them to be as professional as they can and respectful to the clubs they’re at because Kilmarnock and Dundee have been helpful.
“Without mentioning any names, there will possibly be players moving on and that will dictate whether we add anyone else to the squad.
“At the moment we are very satisfied with what we’ve got but if someone wants to leave we probably would have to look for cover.”