In one sense, though, the cascade of foreign crises that now bedevil Mr. Obama could play to Mrs. Clinton’s advantage. By presenting herself in her book and in any possible campaign as the toughest voice in the room during the great debates over war and peace, she could set herself apart from a president who critics charge has forsworn America’s leadership role in the world.

Mrs. Clinton has scrupulously avoided publicly criticizing Mr. Obama; White House aides said he still called her for advice. And much of the administration’s foreign policy still bears her imprint, like the Iran sanctions and a more confrontational stance toward China, which she pioneered and Mr. Obama has embraced.

But in recent interviews, two dozen current and former administration officials, foreign diplomats, friends and outside analysts described Mrs. Clinton as almost always the advocate of the most aggressive actions considered by Mr. Obama’s national security team — and not just in well-documented cases, like the debate over how many additional American troops to send to Afghanistan or the NATO airstrikes in Libya.

Mrs. Clinton’s advocates — a swelling number in Washington, where people are already looking to the next administration — are quick to cite other cases in which she took more hawkish positions than the White House: arguing for funneling weapons to Syrian rebels and for leaving more troops behind in postwar Iraq, and criticizing the results of a 2011 parliamentary election in Russia.

The criticism of the Russian election led Mr. Putin to accuse her of fomenting unrest, and left some senior Obama aides unhappy. “Some at the White House thought she overstepped,” said Michael A. McFaul, a former ambassador to Russia, who supported her view.

At the same time, Mrs. Clinton’s instincts were curbed by her innate caution, her determination to show loyalty to a rival-turned-boss and her growing pains in the job. Still, dissecting her record yields tantalizing clues about what kind of foreign policy she might pursue as president. “Hillary unbound,” people who worked with her say, would be instinctively less reluctant than Mr. Obama to commit the military to foreign conflicts.

“It’s not that she’s quick to use force, but her basic instincts are governed more by the uses of hard power,” said Dennis B. Ross, a former White House aide who played a behind-the-scenes role in opening secret direct talks with Iran about its nuclear program.