Kent impressed on loan at Rangers last season and Steven Gerrard is desperate to bring him back for a second spell.
However Liverpool are currently only interested in a permanent sale with Jurgen Klopp recently ruling out another a loan move.
But with the transfer window in England shutting tomorrow at 5pm time is running out for a move to be completed.
And while Leeds hope to push through a deal it is understood Kent himself would prefer to be at Ibrox next season.
Marcelo Bielsa is keen to seal the winger and could even use him as a forward to replace Kemar Roofe.
But if the deal falls through Rangers will enquire again to sell if there is any chance of bringing him back.
Rangers are at an advantage due to the transfer window being open until September 2 in Scotland and Kent remains their first choice signing.
Gerrard though will move onto other targets if either Leeds seal the signing or Liverpool remain insistent that a loan deal if out of the question.
Rangers are down a winger after letting Daniel Candeias and with Glenn Middleton set to go out on loan.
Shey Ojo and Scott Anfield started in support of Alfredo Morelos on Sunday against Kilmarnock, while Jordan Jones, Jake Hastie and Jamie Murphy are also in contention.
But Kent would be the icing on the cake for Gerrard who has overseen another positive summer of business at Ibrox.
Kent fly out on Liverpool’s pre-season tour but wasn’t involved in the Community Shield last weekend.
Leeds started their Championship with a 3-1 win over Bristol City on Sunday, but Bielsa revealed his frustration with Roofe’s departure.
He said: “I can’t be happy if Roofe leaves.
“If I tell you this when he scored 15 goals last season, I would be stupid.”
The Argentine added: “We have to adapt if it finally happens.”