Barcelona are currently leading the race to sign Griezmann from Atletico Madrid at the end of the season.
The Catalan giants remain on the lookout for a big name signing despite adding Ousmane Dembele to their ranks back in the summer.
Griezmann could move to Barcelona as a replacement for Suarez, who has managed to score five goals from 15 appearances so far.
But Spanish website Diario Gol say Messi does not want the Uruguay international to be sacrificed at the end of the campaign.
And, as a result, he is doing his utmost to stop Barcelona from pressing ahead with their plans to sign Griezmann.
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The France international, like Suarez, has enjoyed a rotten campaign.
Atletico fans turned on him during the Madrid derby at the weekend, with the Wanda Metropolitano crowd booing and jeering him.
And Messi, it appears, has similar doubts.
Diario Gol say the 30-year-old has made it clear to Barcelona president Josep Maria Bartomeu that Suarez’s services must be retained at all costs.
Whether they bow to their talisman’s demands, however, remain to be seen.
One trump card for Messi is that Barcelona may instead focus on Liverpool playmaker Philippe Coutinho.
Barcelona saw three bids for the Brazilian fail during the summer transfer window.
But, with Coutinho struggling for consistency since his move fell through, he may get his wish to play at the Nou Camp granted.