The Daily Mail says in competition with Manchesster United, Newcastle have made their offer this Saturday morning.
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Toon's offer is for £65.5m up-front fee plus an additional £4.4m in add-ons.
Newcastle are now waiting on Leipzig's response and are confident of positive news. However, personal terms with Sesko still need to be discussed.
Newcastle see Sesko as their No1 target to replace Liverpool target Alexander Isak.
United remain keen, though are yet to make an offer themselves.