Blackburn have already enquired about the striker, while Middlesbrough are thought to be lining him up if they fail in a bid for Manchester United forward Alan Smith, who is also interesting Everton.
Rovers had hoped to include Jason Roberts in a cut-price deal for Beattie but, according to the Daily Mail, the Toffees want to recoup more of the £6million they paid for him to fund their Smith swoop.
Rovers were keen on a deal for Beattie during the January transfer window but David Moyes was unwilling to let the 29-year-old leave at the time.
However, with Everton now favourites to land Smith ahead of Boro, Moyes wants the former Southampton star off his wage bill and will be hoping to receive a firm offer this week.
If Gareth Southgate misses out on Smith and Beattie, he may then turn his attention to Tottenham's Mido who is available for £2million.
Spurs are reportedly keen to get rid of a player who is now fifth-choice up front following the recent arrival of Darren Bent.
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