Liverpool chiefs hope to close the signing of Florian Wirtz before the Club World Cup transfer deadline falls on June 10.
While neither Liverpool nor Bayer Leverkusen are in the competition, talks are continuing with the aim of an agreement being struck before the window closes.
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The summer transfer window will again open soon after, but all parties want a deal done sooner rather than later.
The Mirror says Wirtz has hopes of being on Merseyside in the early part of this week to undergo a Liverpool medical and sign his contract.
Liverpool's latest offer is for £100m up front plus £13m in bonuses. However, it still falls short of Bayer's £126m askng price.
Despite the difference, Liverpool sports director Richard Hughes is confident of an agreement this week.