Atletico Madrid star Griezmann was heavily linked with a mega-money switch to Manchester United this summer worth a around £86million.
However, United's interest in the Frenchman has cooled recently, with manager Jose Mourinho preferring to sign a genuine No.9 due to the long-term injury suffered by star striker Zlatan Ibrahimovic.
Griezmann also tweeted last night saying "Now more than ever #Atleti" suggesting he is happy at the La Liga club.
But Giggs believes the deal could rumble on throughout the summer and says a striker is a priority signing for Mourinho.
Ryan Giggs believes Antoine Griezmann's move to Man Utd is not dead yet Manchester United fans suggest Antoine Griezmann alternatives Thu, June 1, 2017
Click to reveal who Man Utd fans want to sign if a deal for Antoine Griezmann doesn't happen
Play slideshow Twitter 1 of 10
Manchester United need goals, all the teams I played in there was goals from everywhere
Man Utd legend Ryan Giggs
"I'm not so sure that we've heard the end of that," he said on Sky Sports.
"I think it will ebb and flow during the summer, there will be lots of rumours.
"Manchester United need goals, all the teams I played in there was goals from everywhere.
Ryan Giggs says Man Utd need to sign a striker this summer
Get Quotes on Home Insurance
"Sometimes you would rely on a centre-forward for a season but more often than not you had midfielders, you even had defenders, you had wingers and obviously the forwards all contributing to scoring goals.
"They did lack goals (this season), there were too dependant on Ibrahimovic and obviously they will miss him at least for the start of the season next year.
"They need to bring players in who are going to score goals and obviously one goalscorer who is going to get 20-plus."