It is believed that the former Spurs and Middlesbrough star, who is currently with the England squad at Euro 2000, is being lined up as part of Leeds manager David O'Leary's summer spending spree as he looks to strengthen his squad ahead of a Champions League campaign next season.
He has already signed Barmby's former Everton colleague Olivier Dacourt and is expected to make a move for Celtic striker Mark Viduka in the coming weeks.
However, a £7.5 million bid for Barmby, whose outstanding form towards the end of last season earned him a late England call, could help O'Leary further improve his midfield and provide Everton manager Walter Smith with more valuable transfer funds.
He is expected to enter further contract talks with Everton when he returns from Euro 2000, although reports claim that his representatives have declined a lucrative new offer.
His current Goodison Park deal has a year left to run before he could be allowed to leave on a Bosman transfer, but Everton officials have been hopeful that he will sign an extension.
Should he reject the offer, though, they could sell him in order to get some return on the £5.75 million which they paid to Middlesbrough in 1996.
