Although no official registry or class was established during the TOS-era when this ship was mentioned, both the Star Trek Concordance and the Star Trek Encyclopedia, 4th ed., vol. 1, p. 262, using the information created by Greg Jein in his The Case of Jonathan Doe Starship article, identified this ship as a Constitution-class heavy cruiser with the registry number NCC-1647. Doug Drexler reiterated this on his Drexfiles blog. In the Star Trek Encyclopedia (4th ed., vol. 1, p. 262), there was a starboard profile pictured of the Farragut as a Constitution-class vessel.
Many other publications, beginning with The Making of Star Trek, p. 165 and Franz Joseph's Star Fleet Technical Manual, listed the Farragut as a Constitution-class heavy cruiser. The Technical Manual. giving the ship a registry of NCC-1702, also listed the vessel as being destroyed, as does The Making of Star Trek, although there was no canon evidence supporting that assertion.