The team doesn’t get to unilaterally decide that a player is going on NFI, especially since NFI players don’t even necessarily get paid. Based on that one Instagram post, it appears that Garcia has been cleared by a doctor, which would be the first step toward arguing against the Patriots putting him on NFI. Obviously, the result of his Jets physical will give us some idea of what’s going on too.
Zach Orr is an example of a player who was cleared by an independent doctor but not by teams. Kyle Love was released after he refused to spend a year on NFI with a medical issue that the Jaguars would later let him play with. AJ McCarron successfully argued (years after the fact) that he should not have been eligible for NFI. Tre Jackson was claimed off waivers by the Rams but then failed his physical with them. Those are all cases that could be somewhat relevant here.