The 42-year-old is now the second favourite with the bookies to move to Ibrox following the sacking of Pedro Caixinha at the end of October.
Rangers have been closely linked with Aberdeen coach Derek McInnes in the last few weeks but caretaker boss Graeme Murty remains in charge for the time being.
He has helped the club pick up six points against Hearts and Partick Thistle so far but van Bronckhorst is now a serious contender for the role.
Reports in the Netherlands claim the former Gers left-back is happy at Feyenoord after he guided them to the league title last season.
However, Feyenoord have struggled both domestically and in Europe this season and the former Netherlands international may be looking for a fresh challenge at the age of 42.