The Yorkshire giants saw a move to sign Bayern Munich midfielder Michael Cuisance collapse on Thursday when the French star failed a medical.
Leeds still want to do business before the transfer window closes and are looking for a replacement target to Cuisance, along with a winger.
They had a bid for Rangers wide-man Ryan Kent rejected earlier in the transfer window and are now focusing their efforts on Leicester.
The Foxes are ready to let Gray depart the King Power Stadium if an offer which they deem acceptable arrives before the window closes.
He has just a year left on his contract at Leicester and is out of favour with boss Brendan Rodgers.
Gray clocked just three Premier League starts under the Northern Irish tactician last season and could be attracted to the idea of a move to Leeds.
With the clock ticking in the transfer window, Leeds have only a limited amount of time in which to do a deal for Gray.
