United have already seen an €80 million bid rejected by the Dutch side, who have told United chiefs they would only consider a financial package worth closer to €100m.
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Sources have told ESPN that United have not given up on signing the Brazilian, partly because he has made it known to Ajax he wants to move to Old Trafford.
The 22-year-old was left out of the squad for the 1-0 win over Sparta Rotterdam on Sunday after informing the club he "did not feel 100% ready to play."
United have informed Ajax they are mulling over making another offer but have also let it be known they have other options as they look to strengthen Erik ten Hag's forward line.
Chelsea, according to sources, are reluctant to send Pulisic to United because of fears they would be strengthening a rival while PSV Eindhoven insist they will not enter talks over a deal for Gakpo until after their Champions League qualifier with Rangers on Wednesday.
Gakpo, 23, represents a cheaper option than Antony although PSV are likely to demand an offer of more than €50m including add-ons.
Meanwhile, sources have told ESPN that United are set to finalise Eric Bailly's loan move to Marseille next week. United have accepted a season-long loan offer for the Ivory Coast defender which will see Marseille pay all of his wages.
Sources have told ESPN the deal also includes an option to buy next summer which would turn into an obligation to make the deal permanent, if certain terms are met such as Bailly playing a certain number of games or Marseille qualifying for the Champions League.