No I think Trump's aides tell him to say it if he wants Alexander to win so they could potentially be on the table in the trade deal. Or he can say 'yes' or 'we're looking into that' and lose Johnson votes so it's not available to US companies.
I think the whole thing is semantics anyway. Technically 'the NHS' wouldn't be on the table. But all the services etc could be put up for tender with the US being given first dibs on them. So although 'the NHS' is still public in terms of funding and availability to all, it's all provided by private companies.
Being a largely closed system it's more liable to profiteering - all you need to ensure is that your tender is the one accepted and we've seen before that hasn't necessarily gone to the cheapest/best available. G4S continually fuck up yet get given huge swathes of govt contracts. If these companies come back and say we can't do it for the quoted price then it's either cough up or let the system fail, which would be catastrophic politically as well as for the population. If it did it would have to be taken back over by the state so once again profits are privatised, losses are nationalised and ends up costing more than if the state had just provided it in the first place.