Bergkamp is widely regarded as one of the greatest players to have graced the Premier League, having scored 120 goals and laid on 71 more during his 423 appearances for Arsenal, helping the club to four FA Cups and three Premier League crowns, including an Invincible 2003-04 campaign.
Grealish has the chance to lead Villa to their first major trophy since 1996 when they take on Manchester City in the EFL Cup final at Wembley on Sunday.
The 24-year-old has been heavily linked with a summer move to Manchester United regardless of whether Villa survive in the Premier League or not, but Pires believes that he would be an ideal replacement for David Silva at Man City.
"Jack is a great footballer with a good technique – and I love watching him play because he reminds me of Dennis Bergkamp. I like to see him playing in the same position as Dennis, as a number 10," Pires told The Mirror.
"He has great vision, he isn't a selfish player because he always wants to do what is best for the team, and I think he has a great future. I don't want to put too much pressure on Jack, because he is still a young player learning the game, and Dennis was an incredible talent and a champion. But Grealish is a very complete player and I think the key for him is to keep playing like he enjoys his football.
"City need to find a replacement for Silva next season - and I think Grealish is someone who Pep [Guardiola] will look at. He is English, he has good experience of the Premier League, and he has a great technique. But I think Grealish is a player who would interest a lot of top clubs, like United , Chelsea and Arsenal.
"This has been a tough season for Villa, but I think Grealish will take a lot from it. He has learned how to fight every weekend and we have all seen how much it means to him to play for the club. Playing against Man City at Wembley is a great stage for Jack. But not just him. This is a chance for Villa to win a big trophy and that would have a big effect on the rest of their season."
Pires played alongside Bergkamp for six years at Arsenal before ending his Premier League career at Villa Park.