The fellow Welshman has urged the 23-year-old to join one of the Liga giants "in the next year or so" so he can fulfil his promise and not run the risk of plateauing
The sought-after winger has been the subject of much praise after scoring three goals in as many games for club and country in recent weeks, with Spurs team-mate Jan Vertonghen claiming the 23-year-old "could play for every team in the world".
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Earnshaw told talkSPORT: "Sometimes you just think that he can be unstoppable and he should, in the next year or so, move to Real Madrid or Barcelona.
"I want to see him at that level and I think that will take him on to a different stage again because he’ll improve again.
"He’s been at Tottenham for quite a while but sometimes it’s nice to see people do well and go higher, and reach the top. There is no higher place than Real Madrid or Barcelona."