The 23-year-old has been heavily linked with a move away from the Eredivisie outfit this summer, with Manchester United believed to be one of the frontrunners for his signature.
A report last week claimed that United have agreed personal terms with the player and his representatives, although the Red Devils are yet to table a formal offer for the Netherlands international.
Gakpo is believed to be one of a number of attacking players on Erik ten Hag's shortlist, but his primary target is Ajax's Antony, with the Brazilian currently in the UK for his medical ahead of completing an £85m switch to Old Trafford.
Gakpo has suggested that his future will be decided within the "next days" and has not ruled out the prospect of staying at PSV beyond Thursday's transfer deadline.
"Antony to Man Utd? Nice for him... but for me and my future, I think that has no influence. It's like a puzzle so in the next days I will see which puzzle fits best - for me, it'd be certainly not bad to stay at PSV. We'll see," Gakpo told ESPN.
Although United seem to be the favourites to land Gakpo should he decide to leave PSV, a number of other clubs have also expressed their interest in signing the Dutchman this summer.
According to The Athletic journalist David Ornstein, Premier League trio Everton, Leeds United and Southampton are among the clubs keen to recruit Gakpo.
However, Ornstein adds that the Saints are the team who have made 'the most concrete approach' so far, and have seen an offer turned down by PSV.
Gakpo has allegedly been identified as a top target for Southampton by Rasmus Ankersen, the co-founder of the club's ownership group Sport Republic, who has led talks with PSV over a deal for the Dutch attacker.
PSV's failure to qualify for this season's Champions League could potentially influence whether Gakpo decides to stay or move on this summer, but head coach Ruud van Nistelrooy insisted last week that he is relaxed over the player's future.
"I think I am very relaxed about [Gakpo's] situation," the former Man United striker told reporters last Tuesday. "There have been rumours about him since the beginning of the season almost eight weeks ago, and we have not seen Cody changing his behaviour up to now.
"There were rumours over the whole period and I admire that a lot in him. He is working on his development on a daily basis, playing in games and reaching his level.
"He is not distracted, he is focused on the games he has to play and focused on tomorrow's game with his teammates. For a young guy like him it is special to see that attitude."
Gakpo, who still has four years remaining on his contract at the Philips Stadion, was a standout performer for PSV last season, scoring 21 goals and providing 15 assists in 46 appearances across all tournaments.
The 6ft 2in attacker has since made a bright start to the new campaign, contributing with four goals and six assists in eight games, including three strikes in as many Eredivisie matches.
Southampton have made six signings so far this summer, recruiting Joe Aribo, Gavin Bazunu, Romeo Lavia, Armel Bella-Kotchap, Sekou Mara and Mateusz Lis for a combined fee in excess of £50m.