The 4th Amendment allows the government to wiretap as long as a warrant has been sought.
The right of the people to be secure in their persons, houses, papers, and effects against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.
With restrictions, the US Government can wiretap. Further, see
Olmstead v. United States
and Katz v. United States
In the Olmstead case, the courts upheld the Federal Government's argument that wiretapping doesn't require a warrant. In Katz, that ruling was overturned.
The NSA case remains at odds with these two cases since FISA is in play.
Private companies are governed by both federal and state privacy laws such as the Electronic Communications Privacy Act (18 U.S.C. 2510 et. seq.) That authority comes from Article 4:
This Constitution and the laws of the United States which shall be made in pursuance thereof; and all treaties made, or shall be made, under the authority of the United States, shall be the supreme law of the land.