Gakpo was high on the Red Devils radar but despite holding talks with compatriot Erik Ten Hag a deal could not be reached and attentions turned to Antony, who moved to Old Trafford from Ajax Amsterdam.
Gakpo was also linked with Southampton and Leeds United in the final days of the summer transfer window but he chose to remain with his boyhood club.
The 23-year-old was particularly keen on the move to Old Trafford and has rued the fact that it failed to materialise.
"I was close to leaving, I spoke to Erik ten Hag a few times at Manchester United," Gakpo told The Times. "In the end, the deal didn't go through.
"[It] was a shame, for me and my development, and because Manchester United is one of the biggest clubs in the world. Also, [it was a shame] for PSV, because to transfer a player to Manchester United is a good thing for the club.
"It ended about a week before the end of the transfer window and in that week I had to decide if I would go to Leeds or Southampton. In the end I stayed, but it was a stressful period."
"I always said if I stay here, it's my childhood club. It's not a punishment," he added. "I like being here, I want to win trophies here and do my best for the club.
"But if I got the chance to go to the Premier League or something, of course I would have to consider that option. At that time, it was hectic for me, but I recovered pretty quickly."
Gakpo has scored nine goals and provided seven assists in 10 Eredivisie games this season.
