Liverpool have rejected three bids from Barcelona for their star Coutinho.
The Reds are determined to keep him despite the player handing in a transfer request.
Coutinho hasn’t played for the club this season – a situation similar to Diego Costa’s at Chelsea, although the Spain international is refusing to even train with the Blues side.
Costa faces six months in exile until he can complete a move to Atletico Madrid and end what has been a poorly-handled transfer saga from both camps.
And while Mills sees no way back for Costa now, he believes Coutinho still has a future at Liverpool if he behaves like David De Gea did when a 2015 transfer from Manchester United to Real Madrid broke down.
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“Klopp is in the huge problem of dealing with Coutinho mentally,” Mills said on 5Live.
“There aren’t too many players that have been refused transfers with the exception of David De Gea.
“I think he’s the only player who had to go back, who was in Madrid and about to sign on the dotted line and whose deal collapsed at the last minute. He came back and his attitude was exceptional.
“Most players if they get their dream move blocked by the club, we’ve seen it with Costa, they go on strike. They don’t quite perform to their ability.
“There’s going to have to be that decision between Klopp and Coutinho, how do they deal with it? How do they build that bridge back?
“To start with Coutinho said he wasn’t interested unless it was by mutual agreement. Now it’s ‘I don’t want to be here anymore’.
“Those bridges have to be built first and foremost.”