The 17-year-old rejected the chance to extend his stay with the Gers and it’s understood the Light Blues fear losing the London-born forward to an English club.
Scotland Under-21 star Gilmour quit Rangers last year and joined Chelsea, and a number of English clubs are believed to be keeping tabs on Mebude’s situation.
Mebude has already been capped by Scotland Under-17s, and is one of the most highly rated products of the Rangers academy, and the Ibrox side are keen to keep hold of the talent.
He signed his first professional deal last May, but that agreement runs out in summer 2019. The club has already attempted to tie him down on a new, improved deal but Mebude has so far rejected all offers, sparking fears he could leave the club in the near future.
Elsewhere, Steven Gerrard could return to Merseyside for a signing target - but on the other side of Stanley Park. Reports suggest Everton midfielder Kieran Dowell is high on the former Liverpool captain’s shopping list, with Gerrard having taken in Scotland Under-21s’ match with their England counterparts at Tyencastle last month, with Dowell impressing for the visitors.
Dowell has made just two appearances for the Toffees this season after a fruitful loan spell at Nottingham Forest last season produced nine goals from 38 appearances in the No.10 position.
Scott Arfield tends to occupy that spot in the Rangers starting line-up but Gerrard’s other options - such as Lassana Coulibaly, Ryan Jack, Ovie Ejaria - aren’t suited to the role and so the Gers boss could look elsewhere in January.
The rumoured pursuit of Dowell could spell the end for Graham Dorrans, while fellow midfielder Liam Burt, who is on loan at Alloa, could return to Ibrox due to a lack of game time at the Indodrill Stadium.
On Monday afternoon Borna Barisic, Nikola Katic and Lee Wallace were all involved as the Gers reserves put five past their kilmarnock counterparts without reply.
A strong Rangers team saw Katic and Wallace start in central defence while Stephen Kelly, who has been involved with the first team squad, also featured.
Jordan Houston scored an early penalty before strikes from Mebude and Andrew Dallas put Rangers 3-0 up. Dallas then doubled his tally with the fourth with Josh McPake setting him up for his hat-trick and Rangers’ fifth.
Barisic was replaced by Daniel Finlayson at the halfway point while Katic made way with 20 minutes left for Matthew Shiels.