The Europa Conference League winners were understood to have agreed a £14m fee with the Citizens to sign the 19-year-old attacker last month.
However, Sky Sports News reports that Borges's move to West Ham has been called off and the teenager is now on the verge of joining Ajax.
The report claims that there were delays in getting the deal over the line for the Hammers due to agent fees and personal terms.
Borges is now said to be heading to Amsterdam for his medical after Ajax swooped in to agree a deal in principle with Man City to sign the Portugal Under-19 international for €20m (£17.2m).
As part of the deal, Man City are understood to have inserted a buy-back option and a 20% sell-on clause.
Borges joined the Citizens' youth system from Sporting Lisbon in 2015 and has since developed into one of the club's brightest prospects, scoring 55 goals and registering 42 assists in 79 appearances from Under-19 to Under-21 level.
The versatile attacker, who primarily operates on the left wing, was named 2022-23 Premier League 2 Player of the Season after scoring 21 goals in 24 games to win the Golden Boot and help Man City Under-21s win their third successive title.
Borges has trained numerous times with Pep Guardiola's first-team squad, but he is now set to depart the Citizens having failed to make a single senior appearance for the club.
West Ham have been left frustrated in the transfer market so far this summer and they are the only Premier League side who are yet to make a single addition ahead of the new season.
David Moyes is keen to spent the £105m recouped from the sale of Declan Rice to
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