The Sunday Mirror says Kelly has rocked Liverpool by turning down a new deal worth up to £35,000 a week.
Raheem Sterling has already rejected a £50,000-a-week offer.
Now defender Kelly is also unhappy. He broke into the England squad last season, but has been sidelined this term by a knee ligament injury.
He has 18 months left on his deal but the Kop are finding salaries earned by big-money buys like Stewart Downing are hampering talks with other players.