Ole Gunnar Solskjaer is looking to bolster his attacking options before the English transfer window closes on Thursday.
However, United will have to return with an improved bid if they are to prise the Brazil international away from the Johan Cruyff Arena.
Ajax boss Erik ten Hag has made Neres integral to his plans for the 2019/20 campaign, and the club are hopeful of persuading him to sign a new contract.
It has previously been reported that Ajax valued the winger at £64m, but the asking price may have risen with the new Eredivisie season just around the corner.
Neres scored 12 goals in all competitions last term as Ten Hag's men won a domestic double and reached the Champions League semi-finals.
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