The Atletico Madrid and France forward has been strongly linked with a move to Old Trafford, with United manager Jose Mourinho insisting he wants quality over quantity in the transfer market.
Goal understands that United are confident of securing a deal even if they fail to qualify for the Champions League this summer, with a deal likely to cost around €80 million.
And Petit, who played in the Premier League for Arsenal and Chelsea, believes Griezmann's bond with Les Bleus team-mate Paul Pogba will help the Premier League side to persuade the 25-year-old to join.
"I think he will come to Manchester United - definitely," Petit said to Sky Sports.
"He has a great partnership with Pogba in the national team, they are very good friends as well.
"He will bring goals, assists and his winning mentality."
Petit believes Griezmann - who only finished behind Cristiano Ronaldo and Lionel Messi in the 2016 Ballon d'Or - would represent a huge steal for United if the deal goes ahead.
He continued: "Griezmann, over the last four years, has improved step by step.
"Every single year he has improved his level all the time so he has got big ambition personally but as well with the team he is playing for.
"For me it would be a big bargain for Manchester United."
Griezmann was a runner-up in the Champions League and Euro 2016 last season and has only won the Supercopa de Espana during his time at Atleti.