Senator Rand Paul of Kentucky, for example, has argued that his noninterventionist outlook on foreign policy would offer unique advantages in a head-to-head race against Mrs. Clinton. His argument: by 2016, Mrs. Clinton will be viewed as a champion of American military action abroad, alienating younger voters of both parties exhausted by a decade of wars. Given the hawkishness of his likely Republican rivals, he alone, Mr. Paul says, can appeal to such disaffected youth.

It is a message Mr. Paul has delivered repeatedly, to the likes of David and Charles Koch, the billionaire conservative industrialists, according to a person familiar with their conversations.

Mr. Cruz takes an entirely different approach, telling donors that Mrs. Clinton’s reputation as a moderate, and one who can appeal to elements of the Republican Party, necessitates the selection of a true conservative like himself. He says his brand of raw, unapologetic right-wing politics and policy can excite conservative voters long frustrated, in his telling, by the Republican Party’s tendency to nominate ideologically bland, watered-down figures, like former Gov. Mitt Romney of Massachusetts and Senator John McCain of Arizona.

Image Gov. Chris Christie of New Jersey Credit... Mel Evans/Associated Press

His argument: moderate Republicans rarely win the White House, and their chances would diminish still further in a race against Mrs. Clinton.

Republicans eyeing the White House are eager to diagnose Mrs. Clinton’s liabilities and shortcomings, however real or imagined. The biggest of them, they contend, is her deep connection to the Obama administration as secretary of state.

At a dinner for wealthy donors last week in Texas, a guest said, Mr. Perry predicted that Mrs. Clinton would become ensnared in the “Barack Triangle” — a play on the Bermuda Triangle — and was indelibly linked to what Mr. Perry said was the president’s mixed economic record, foreign policy struggles and detached governing style. Mr. Cruz, latching on to the same theme, has begun referring to the “Clinton-Obama” agenda.