Barkley, who is injured at the moment, is out of contract at Everton at the end of the season.
According to The Liverpool Echo, London clubs Chelsea and Tottenham Hotspur wanted to sign the attacking midfielder in the summer transfer window.
A report in The Telegraph in October claimed that Spurs manager Mauricio Pochettino wants to recruit the 24-year-old in the January transfer window.
Everton manager Allardyce has suggested that he is not sure what will happen with Barkley regarding his future.
Allardyce told The Liverpool Echo when asked if Everton should be looking to keep Barkley: “Well they’ve tried haven’t they?
“They’ve tried and they’ve tried and there’s only so far a football club can go but I’m not privy to the exact details.
“But the most important thing for me and for Ross is not to worry about a contract, it’s to get fit and perform at whatever level he performed to before.
“Because if he stays here that will be great, if he goes somewhere else he will need to hit that level to get the contract he wants.”
Barkley is a very good attacking midfielder, and he would be a good addition to the team once he recovers.
With a new manager in charge of the Toffees now, the England international should be doing all he can to impress him and re-establish himself in the Everton starting lineup.
The Toffees are in danger of getting relegated to the Championship at the end of the season.