Naval Vessels Transfer Act of 2013 - Authorizes the President to transfer on a grant basis to: (1) Mexico, the OLIVER HAZARD PERRY class guided missile frigates CURTS and MCCLUSKY; and (2) Thailand, the OLIVER HAZARD PERRY class guided missile frigates RENTZ and VANDEGRIFT.

Authorizes the President to transfer on a sale basis the OLIVER HAZARD PERRY class guided missile frigates TAYLOR, GARY, CARR, and ELROD to the Taipei Economic and Cultural Representative Office of the United States (which is the Taiwan instrumentality designated pursuant to the Taiwan Relations Act).

States that: (1) the value of such vessels transferred on a grant basis shall not be counted against the aggregate value of excess defense articles transferred to countries in any fiscal year under the Foreign Assistance Act of 1961; (2) transfer costs shall be charged to the recipient; and (3) to the maximum extent practicable, the country to which a vessel is transferred shall have necessary vessel repair and refurbishment carried out at U.S. shipyards (including U.S. Navy shipyards).

Terminates transfer authority three years after enactment of this Act.