Not sure about that. The USAF should really set a minimum spec and go for lowest through-life cost - Boeing has courted Saab for that, not for the Gripen design per se.
On the other hand, the USAF uses T-38s and the Navy uses F-5s for a lot of non-training things, like companion aircraft for expensive F-22s and as aggressors. If I was writing the T-X requirement I would not let those things drive me to a supersonic T-X, but I might consider what I could do with a low-cost, modern fighter as a lead-in/aggressor/companion.
And if I was Boeing-Saab I would be sniffing around Guard and Reserve units and pointing out that it might be a long time before they get F-35s, and wouldn't they rather be doing air defense and CAS with a hot new fighter rather than flying UAVs?
Peter We - T-X is not a T-45 replacement and there is no requirement that it be off-the-shelf. Even the "existing airframe" designs will (according to all precedent) end up being highly customized.