Everton midfielder Barkley has been told if he does not sign a new contract he will be sold.
The England international’s current deal expires in 2018 and Everton don’t want to lose him on a free transfer next summer.
And the Daily Star Sunday claim Barkley could be heading to north London in a £30million deal.
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According to the report, Arsenal manager Arsene Wenger sees Barkley as the next big thing at the Emirates and is keen on securing a deal.
Express Sport understands Wenger will extend his stay at Arsenal into next season – despite calls from some fans for him to leave.
The futures of Mesut Ozil and Alexis Sanchez are in doubt, however, while Jack Wilshere is tipped to be offloaded in the summer, leaving room for Barkley to come in.
Arsenal and Tottenham want to sign Everton's Ross Barkley
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Barkley and Romelu Lukaku could both leave Everton this summer
Barkley will reportedly cost around £30m
However, Tottenham have plans to launch a rival bid for the Goodison Park star as they get ready for a season at Wembley.
Spurs are yet to announce their year-long deal to host home games in west London.
But manager Mauricio Pochettino wants to settled team in place for their move across the capital.
Everton already face losing Romelu Lukaku in the transfer window this summer, with the Belgium international keen on a return to Chelsea.