Or they'll take it to the courts. But most likely they'll reach an agreement on a set fee with Man City, which suits all parties.
Going to court makes no sense as it would never be resolved in a timeframe that would allow him to make the transfer before the window closes. He'd 100% be better off just playing this season and going to City next year on a free and avoid the litigation mess with Barca. It's a terrible look on both parties.
Also the fact that Jorge Messi is trying to 'negotiate' his exit (despite Messi indicating he is free to leave) with Barto is indication that they know a lawsuit would ruin everything.
Messi needs to tell City to make an offer and he also needs to tell Barca that if they don't negotiate his transfer this window then he is leaving on a free in 9 months.
It's the only way it's going to work out for all 3 parties.