Van Bronckhorst, who took over from Gerrard, has not used Lundstram much since arriving at Ibrox and his former Blades boss Chris Wilder is reportedly looking reunite with him.
The Daily Record claims that Middlesbrough, managed by Wilder, is “monitoring the situation” for Lundstram, who signed a three-year deal with Rangers in the summer.
Ryan Jack is due to return from injury next week and Van Bronckhorst is understood to be a big fan of the Scotland internationalist, who plays in the same defensive midfield role as Lundstram.
Van Bronckhorst also wants to bring in some of his own midfielders, but may have to offload some current players off the wage bill before doing so.
Rangers are also looking to offload central defender Jack Simpson, another player who has not featured much under Van Bronckhorst.