The Brazil attacker is eager to return to Betis after last season's successful loan.
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However, while Betis and Fajardo are pushing for a new loan arrangement, United are insisting on a straight sale and at terms that are out of reach of the Seville club.
Ahead of their 1-1 draw with Elche in the LaLiga's opening round, Fajardo said of Antony: "We have to be respectful of the situation.
"Maintain caution, in line with what we've been discussing. It's a complex operation, given the profile of the player he is, and the profile of the club he belongs to.
"I don't like speculation in the middle of the transfer window."