The 16-year-old has attracted attention from a host of different clubs after impressing in his breakthrough season in the Eredivisie.
Gunners boss Arsene Wenger has been credited with an interest in bringing the talented teenager to the Emirates Stadium, as he strives to secure some of Europe's brightest young talent.
However, Arsenal will face competition from big-spending City, with Roberto Mancini equally keen to commit Bazoer's long-term future.
PSV are determined to keep him out of the clutches of the Premier League sharks that are circling, though, and offered him a three-year deal at the turn of his 16th birthday on October 12, which the player is yet to sign.
The club's sporting director, Marcel Brands, has urged Bazoer to stay, warning him that there will be no guarantee of first-team football in England.
"We have offered him a professional contract now he has turned 16. I speak with the family, the boy and his agents. We hope we will get an answer soon," he said.
"What is his intention? Is it to develop or go for bigger money in England? You have to ask him and his agent which clubs are interested in him. I don't know and it doesn't matter.
"It is his choice and the agent will do their best for the player. The only thing we can do is offer him good possibilities in terms of coaching and prospects."
Rules in Holland prevent Eredivisie clubs offering players under the age of 18, contracts longer than three years, and with FIFA's £296,000 compensation cap for academy players, Brands conceded that PSV cannot compete with rivals in England, Spain and Italy.
"Nobody can. It's not a financial thing, anyway. It's important what is possible in a sporting way.
"It's difficult in England to reach the first team of the big clubs after joining out of the youth academy. A lot of young players from Holland went to England and have not yet played in the first team. Many of them ended up being loaned out to other clubs.
"The players who were successful in England came later, like [Edwin] van der Sar, [Jaap] Stam and [Dennis] Bergkamp.
"PSV are delivering a lot of young players for the European market in Spain, England and Italy. It is difficult to compete with those kinds of clubs. The only special thing we have is that a lot of young players play in the league in Holland."