The Constitution allows the US Congress to generate “letters of marque and reprisal” to allow specific entities (traditionally a specific ship but not limited to that) against a specific enemy. This allowed the Congress to authorize private warships to attack enemy vessels and it was done under the authority of the Constitution. It meant they were operating under the rules of warfare, rather than as pirates.
There is nothing in the Constitution that would disallow them from generating letters of marque and reprisal against Al Qaida and its allies, as enemies of the US, and thence creating the authority for the NSA to go after any and all of them wherever they are in the world. This would authorize the NSA to record all of the conversations that are had by ANY immigrants in the entire country as part of that mission.
It would still not do anything about the unconstitutional and illegal activities that have been going on regarding government collection of data on US citizens. The Constitution says that we are to be secure in our papers and effects. This means that the government cannot open our mail without a warrant. It does not take a leap of genius to conclude that they cannot take our electronic mail without the same standard of requiring a warrant, no matter HOW many FISA court rubber stamps they get.