The boyhood Evertonian was linked with a move to Tottenham Hotspur this week.
The London Evening Standard reported on Wednesday that Tottenham Hotspur had taken a liking to Barkley and lodged a tentative enquiry as a result.
However, the Sun's report states that there has been no approach yet for the England midfielder, who it is claimed has lost the patience of Koeman.
Barkley has 18 months remaining on his Everton contract and is in danger of leaving Goodison Park for nothing if a new deal cannot be agreed before then.
The Sun speculates that Barkley's "difficult relationship" with his manager, who restored Barkley for Tuesday's win against Arsenal, could be a factor were negotiations to commence.
The London Evening Standard reported that Everton will demand £35 million for Barkley this winter, with the Toffees' asking price dropping to £25 million next summer according to the Telegraph - and his valuation will only decrease as he enters the final 12 months of his contract.
A move Tottenham, whose squad comprises several previously untrusted Englishmen groomed into full internationals by manager Mauricio Pochettino, would likely appeal to the Liverpudlian.
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