No, I think teams have 8 spots for international players (could be wrong) but they could use mechanisms like TAM/GAM (read in article below) to trade for more international spots if needed from other teams (and there are always teams who "tank" and/or don't want to spend money on foreign stars) so player's nationality is not really a problem for most teams.
What you've probably meant is they have 3 DP (designated players) spots but there are other mechanisms (TAM, GAM) which will allow them to bring Alba, not sure about Suarez though unless he'll accept really low wages (like a "normal" player) to play with his amigos again (he probably will tbh).
Here is a good explanation.
Gregore, with a salary of $826,000 according to the MLS Players Association, will have his contract bought down using general allocation money (GAM), one of the league's roster-building mechanisms. That will clear a path for Busquets to sign a DP deal of his own.
For Alba, it's both simpler and more complicated.
He is expected to join using targeted allocation money (TAM), which caps his salary at $1,612,500 per year. Every team in the league has $2,720,000 in TAM available to it in 2023 -- essentially supplementing the salary cap -- but as part of the sanctions levied against Inter for the past violations, the club has had $2,271,250 deducted from its allocation allotment spread across the 2022 and 2023 seasons, making it a scarce resource on South Beach.
Miami employed salary cap gymnastics to sign Lionel Messi & Co. If MLS wants to be a destination league, it must cater to its most ambitious clubs.
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