The France international has announced that he will leave Spanish side in the summer transfer window after five seasons with the club.
Griezmann is thought to be available in a cut-price £108m deal if a suitor meets the clause in his contract before the deadline on 30 June.
FC Barcelona are thought to be keen on a move to sign Griezmann after their 4-3 aggregate loss to Liverpool FC in the Champions League semi-finals.
However, ESPN pundit Burley believes Liverpool FC should consider an audacious bid to sign the 28-year-old striker this summer.
“I actually think that Liverpool would be a great place for Griezmann to go this summer,” Burley told ESPN FC.
“Liverpool are very much a team on the up at the moment and he could aid Roberto Firmino who also likes to drop deep.
“Liverpool are a huge club who are fighting for league titles with Manchester City and have reached the last two Champions League finals.
“I don’t know if they have got enough money to do it but I wouldn’t go to Manchester United, that’s for sure.
“You talk about how good Griezmann would be linking up with Messi and Barcelona, what about with Liverpool’s front three?
“You look at that and we talk about Liverpool needing that strength in depth, he would add that and Firmino or Salah some games might sit out.”
Liverpool FC finished in second place in the Premier League table this season and a point behind champions Manchester City.
The Reds can still end the season with silverware if the Merseyside outfit manage to beat Tottenham Hotspur in the Champions League final on Saturday 1 June.
Jurgen Klopp lost Roberto Firmino to injury for their final three Premier League games to highlight their lack of depth in forward positions.
Liverpool FC have never won the Premier League title but have finished in second place on four occasions.
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