Celtic had agreed a £1m deal with Everton for Vaughan, 22, with an initial loan fee in place but Burnley manager Eddie Howe is keen.
Loan move: Vaughan
Eagles is an option for Everton but Burnley's asking price had put them off.
They have also been exploring loan deals as David Moyes has no cash to spend. Tottenham, meanwhile, remain interested in Everton captain Phil Neville but consider a deal difficult following the purchase of Steven Pienaar.
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