The 36-year-old forward is expected be reunited with his ex-colleague and close friend in South Florida this summer.
Suarez, who joined Gremio earlier in 2023 on a two-year deal, is ready for one last hurrah with World Cup winner Messi Stateside.
According to ESPN Brasil, the Brazilian club are aware of the striker's intentions of wanting to leave.
Gremio are likely to sanction a move to allow Suarez to leave, which would see him join Messi, Sergio Busquets and Jordi Alba at Miami.
Suarez is understood to be prepared to fund the move himself, by negotiating a settlement, repaying all the wages he received in 2023.
READ MORE ON LIONEL MESSI
Argentina star shuns Messi and showcases Ronaldo tattoo at Women's World Cup
'Absolutely it's possible' - Messi can guide Inter Miami to MLS glory despite sitting last
It is believed a fee of around $10m [£7.8m] will be enough to remove him from his current contract this month.
Gremio coach Renato Gaucho has since labelled the situation a 'problem', with Miami having already made room for the Uruguayan's arrival.
The club have purchased an extra international slot from Western Conference side San Jose Earthquakes in exchange for $150,000 [£117k] in 2023 General Allocation Money (GAM).
Since joining Gremio in January on a free transfer, Suarez has netted 16 goals and assisted nine more in 31 matches.
Most read in Football
World record Mbappe bid, Man United Kane move unlikely, Zaha set for Turkey move
Chelsea transfer news LIVE: Wahi deal advanced, Mbappe latest, Gallagher exit, Olise bid
Darren Ambrose lost work over harsh views on Daniel Levy and Joe Lewis
Adrian concedes three goals in 30 minutes as Liverpool draw to second tier German side
"On the 2nd [of August when the transfer window closes in Brazil], this soap opera will come to an end," Gremio coach Gaucho said after his side's 1-0 defeat to Atletico Mineiro.
Suarez, who watched the loss from the stands, has been out of action since May, nursing a knee injury.