I'm not sure of what you're getting at with this question. I'm assuming that the money was already appropriated by Congress, but for some reason hadn't yet made it to the field, and that that was what DeLay was inquiring about.
Clark said the money was on the way yet we know that Congress has not appropriated any money for this, tho they say that they will. We have to assume that this "money on the way" is a grant because Operation Linebacker kicked off with grant money.
Once you start down this path of the locals taking care of it, your right back to the problem of the feds re-imbursing the state/local for these costs. The record on this is not too good.
And even if that money is forthcoming, these local jurisdictions don't have a significant amount of space to hold detainees so your back to the feds picking up the detainees when they are supposed to.
The reality is that it is going to take a lot of money to change the philosophy of law enforcement. Delay and other Congressmen know this. Everybody likes dog and pony shows.