The Daily Star Sunday says Manchester City have joined United in hoping to prise Barkley, 20, from Goodison.
United boss David Moyes gave the midfielder his debut when he was in charge at Everton and has made him a top target.
Moyes and United chief executive Ed Woodward had a five-hour meeting last Monday to discuss transfer targets and Barkley was a key part of the conversation.
The United boss would like the deal done BEFORE next summer’s World Cup.
Txiki Begiristain, City’s sporting director, also claims to have been tracking Barkley for two years.
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