The talented defender will be in a position to sign his first professional deal when he turns 16 later this month and Chelsea and Liverpool are also monitoring the situation, according to The Guardian.
Bogarde is rated as one of the most promising young players in Europe and Pep Guardiola, the Man City manager, is said to see similarities between him and John Stones, who he signed from Everton two years ago.
Manchester City, who wrapped up the Premier League title last month, have invited Bogarde and his family to visit their Academy but United and Arsenal have also tabled similar proposals.
United will hope good relations with Feyenoord puts them in a better position than their domestic rivals – the club signed Tahith Chong from the Dutch side in 2016.
With a number of Premier League clubs interested in securing his signature, Bogarde is expected to decide on his next destination after considering which team will give him the best chance of eventually progressing to the first-team.
Manchester United, Arsenal and Liverpool are all believed to have scouted the youngster extensively this season, while RB Leipzig are also known admirers.
The players uncle Winston earned 20 international caps with Netherlands and had a spell in the Premier League with Chelsea.