The Red Devils have already seen an offer of £12 million rejected by the Goodison Park outfit, but Moyes has prepared a fresh offer that is likely to tempt Everton into parting with one of their most prized assets.
Everton boss Roberto Martinez has stepped up his efforts to find a replacement for the United-bound Baines, giving the clearest indication yet that Moyes is close to luring his former charge to Old Trafford.
Manchester United have also been heavily linked with Baines' teammate Marouane Fellaini, but that deal is not thought to be imminent, though Moyes may well step up his efforts to land the Belgian international should his pursuit of Barcelona's Cesc Fabregas come to nothing.
Despite quotes from the ex-Arsenal captain saying he wants to stay at the Nou Camp, United are aware that Fabregas isn't confident of securing regular first-team football with the Catalan giants, and an increased bid of £40 million may be enough to bring Fabregas to Manchester.
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