Barcelona hired former player Koeman as their new head coach on Wednesday after the sacking of Quique Setien.
The Catalan club plan major changes at all levels following their humiliating 8-2 defeat to Bayern Munich in the Champions League.
Liverpool have yet to agree a new contract with 29-year-old Wijnaldum, whose deal runs out next June.
Although Barcelona have already added Miralem Pjanic, 30, to their midfield, Koeman has eyes on Wijnaldum.
According to AD, citing sources from the player’s camp, Koeman has asked Wijnaldum not to renew his contract.
With just a year remaining on his deal, that makes Premier League winner Wijnaldum an affordable option for Barca.
If he was to swap Anfield for Camp Nou, he would link up with another Dutch midfielder, Frenkie de Jong.
Wijnaldum made 47 appearances for Liverpool in their title-winning campaign, scoring six times.