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The newspaper has revealed that a week ago Carroll was told by "a senior international figure" that he was "still on England's radar for the summer tournament".
However, it is understood that the forward needs to score a few more goals for his club to prove he deserves a place on the plane.
Carroll and the rest of his team-mates travel to Newcastle on Sunday as they prepare for a showdown that could decide who takes a Europa League place.
The £35 million man still has question marks hanging over his Anfield career. He left Newcastle in January 2011 but has so far netted only eight goals in 47 games for the Reds.
However, with Wayne Rooney suspended for the first two games of Euro 2012, Darren Bent injured and Daniel Sturridge and Danny Welbeck both lacking experience at full international level, England are somewhat short of striking options, so 23-year-old Carroll still has an outside chance of making the Three Lions squad.