The Whites are understood to be keen on the Wales international as Marcelo Bielsa continues to whip his squad into shape.
Wilson is thought to be available this summer, but only on a permanent deal according to the Liverpool Echo, having been shipped out on loan for most of his senior career.
Newcastle, Southampton and Leeds are believed to have made an approach for Wilson in this transfer window, with a figure of £12.5m floating about.
The Sun report that Leeds are willing to increase their bid to £15m to secure the 23-year-old’s services.
But Daily Star Sport understands that Liverpool may wait until after their clash with Leeds on Saturday to allow the move to go through.
As a back up, Leeds are also keen on Rangers and ex-Liverpool winger Ryan Kent.
But while Ibrox boss Steven Gerrard issued a firm “hands-off” warning to Bielsa, he admitted that the situation could be out of his hands.
The Anfield legend said: "The message to Leeds is the opening bid is nowhere near the valuation and the player doesn’t really have one as we don’t want to sell.
“We want the player to stay here, as do the board. The fans love him also and Leeds will need to respect that.
“They really need to think hard if they want to come in again. The message is ‘hands off’.
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“What I will say is that if the valuation of any player is met then, as much as I would like the final say on it, it will probably be taken out of my hands.
“To be honest, I’m not really interested in reading into it whether they are going to come back or not.
"For me, it’s just a strong message that he’s not for sale.”