The Wales international was put up for sale by Liverpool during the international transfer window, but the Reds failed to find any clubs willing to match their £20m valuation.
Burnley held talks with Liverpool in an effort to land Wilson, but were unable to reach an agreement with the champions.
Liverpool ended up keeping the 23-year-old beyond last Monday’s international transfer deadline, resisting a loan departure for the player.
However, with the domestic window open until Friday, Wilson is now expected to move on loan and Swansea are in the driving seat.
It is claimed that Liverpool were impressed with the way Swansea handled Brewster on a loan spell last season.
Brewster returned to Liverpool and was sold, and if Wilson impresses at Swansea he could also be sold next summer
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Wilson spent last season on loan at Bournemouth and could come up against the Cherries in the Championship if he moves to Swansea.
