Twenty-second Amendment to the United States Constitution
Some argue that the 22nd Amendment and 12th Amendment bar any two-term president from later serving as vice president as well as from succeeding to the presidency from any point in the presidential line of succession. Others contend that the original intent of the 12th Amendment concerns qualification for "service" (age, residence, and citizenship), while the 22nd Amendment, concerns qualifications for "election", and thus (strictly applying the text) a former two-term president is still eligible to "serve" as vice president (neither amendment restricts the number of times an individual can be elected to the vice presidency), and then succeed to the presidency to serve out the balance of the term (though prohibited from running for election to an additional term).
Twelfth Amendment to the United States Constitution
The Twelfth Amendment explicitly states the also apply to being Vice President. It is unclear whether a two-term president could later serve as Vice President. Some argue that the Twenty-second Amendment and Twelfth Amendment both bar any two-term president from later serving as Vice President as well as from succeeding to the presidency from any point in the United States presidential line of succession. Others contend that the Twelfth Amendment concerns qualification for "service", while the Twenty-second Amendment concerns qualifications for "election", and thus a former two-term president is still eligible to "serve" as Vice President. The applicability of this distinction has not been tested, as no former president has ever sought the vice presidency, and thus the courts have never been required to make a judgment regarding the matter. Hillary Clinton said during her 2016 candidacy that she considered naming Bill Clinton as her Vice President, but was advised it would be unconstitutional.