Beale, who managed Rangers for a total of 10 months, parted ways with the club on Sunday after three defeats in their first seven league matches this season.
Lampard was most recently the caretaker manager of Chelsea between April and May this year and recorded just one win in his 11 matches in charge before the arrival of Mauricio Pochettino.
With the Glasgow-based club currently looking for a new manager, Lampard is keen on discussing a potential deal at Ibrox, according to BBC Sport.
The former England international is without a club after he left Chelsea at the end of last season, when the club opted not to extend his interim deal.
Rangers currently have Steven Davis and Alex Rae serving as the interim management team.
Lampard began his managerial career with Derby County in 2018 and held a permanent role at The Blues from July 2019 to January 2021.
The 45-year-old also spent 44 games in charge of Everton and helped steer The Toffees away from the threat of relegation but was sacked at the start of this year after a ninth loss in 12 league matches.