Jones, 25, emerged as Rafa Benitez' top transfer target during the January transfer window but the Spaniard failed in his attempts to land the Trinidad and Tobago star.
Furious Black Cats boss Steve Bruce foiled Liverpool's loan move by slapping a £15million price tag on the towering forward. But Jones has not given up hope of securing a big-money switch to Merseyside.
Red alert: Kenwyne Jones has not ruled out a summer switch to Liverpool
'If one of the world's big clubs comes in for you, it's understandable a player might have the desire to go to that next level,' he said.
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