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).
Twenty-second Amendment to the United States Constitution
The practical applicability of this distinction has not been tested, as no former president has ever sought the vice presidency, although in 1980, former president Gerald Ford was mentioned as a possible vice presidential running mate for Republican presidential nominee Ronald Reagan, and there were some negotiations between the two camps. Nothing ever became of the idea. During Hillary Clinton's 2016 presidential campaign, she jokingly said that she had considered naming her husband Bill Clinton as her vice presidential running mate, but had been advised it would be unconstitutional. Most likely, the constitutional question raised will remain unanswered unless the situation actually occurs.