In the spring of last year, Timothy F. Geithner wanted to leave his job. The Treasury secretary’s family was moving to New York for his son’s senior year in high school, and the commute to see them each weekend was sure to be arduous.

Who could do his job? Geithner’s answer was Secretary of State Hillary Rodham Clinton.

He ultimately failed to make the case, but it wasn’t for lack of trying. The effort illuminated one of the more intriguing relationships in Washington — the evolving rapport between President Obama and his high-wattage secretary of state.

Before he could depart, Geithner had to convince the White House. He needed to find someone who could glide through Senate confirmation, comprehend the growing debt crises, here and abroad, and communicate administration goals in last summer’s debt-ceiling debate to Congress, to Wall Street and to the world.

The talks gained traction inside the White House before being scuttled shortly before the debt-ceiling talks on Capitol Hill intensified, according to one former and two current administration officials.

The Cabinet shuffle made sense, according to the officials, because Clinton could fulfill all those requirements. Plus, she was believed to be ready for a break. She had described in public her travel-heavy job as “grueling.”

Geithner had submitted a list of names to the White House. Chief of Staff William Daley appeared to “slow-walk” and rob the Clinton suggestion of any momentum, according to one of the administration officials. But actually, Daley was conducting his own vetting process, another official said.

He broached the idea with Clinton. An administration official familiar with the exchanges characterized her response as “cautious interest.” A person close to Clinton had a different take: “She listened respectfully and politely.”

Daley called a few trusted eminences on Wall Street, sounding them out on the personnel switch. Their response was resoundingly positive, both officials said. She had never been a banker, but as a senator from New York, Clinton had cultivated many relationships within the financial sector. Some of them had been longing for the kind of attention they had received from her and her husband, former president Bill Clinton, but rarely got from Obama.

And unlike Geithner, who disdained high jinks on Capitol Hill, Clinton had an intuition for political risk. She knew committee chairs. As the debt crisis worsened in the United States and Europe, Clinton’s popularity abroad would have also allowed her to talk sense to other leaders.

Here and overseas, her appointment would have been “a really positive headline,” said one of the officials.

A spokeswoman for the Treasury Department declined to respond to an inquiry on “private recommendations from the secretary to the president.” The White House said it does not comment on internal personnel discussions.

Philippe Reines, a longtime aide of Clinton and current deputy assistant secretary of state, said in an e-mail: “Without getting into private conversations, it’s been crystal clear from her very first day here that Secretary Clinton is fiercely committed to this Department.”

Clinton and Obama, despite what many forecast, had developed a “humorous rapport,” in the words of one of the administration officials. If she were chosen for Treasury secretary, it would reinforce that the relationship had not only healed but thrived. And after she contributed to many diplomatic and military successes, Clinton would be taking over the most crucial and fraught task of Obama’s presidency: rescuing the economy.

Despite their rivalry on the campaign trail in 2008, the president welcomed Clinton into his administration, acknowledging that her experience would be crucial to his goals.

But some within the administration remembered the difficulties once Clinton came on board. Obama had guaranteed Clinton that she could pick her own team at State when he first negotiated with her in December 2008. And while the principals grew fond of each other, it took more time for their loyalists to follow suit, several administration officials said. Animosity lingered as the White House squared off with the State Department over many nominees.

“To describe it like an Israeli-Palestinian tension was pretty close,” the official said.

That drama has thoroughly subsided. Yet as the White House deliberated the job change for Clinton, a concern arose, according to one of the administration officials. Some of her aides would certainly follow her from Foggy Bottom to the building next to the White House. And “staffing up all over again” could revive the discord.

That problem, among others, scuttled Geithner’s proposed successor. There was another major obstacle: Obama didn’t want him to go. Geithner agreed to stay put, and, to help on the home front, the president talked in person to Geithner’s wife. Obama let Carole Geithner, a social worker and novelist, know that as much as she wanted her husband home for their son’s senior year, the president needed Geithner in Washington.

Shortly after the idea lost currency, Geithner appeared onstage in Chicago for a dialogue with Bill Clinton. At the closing session of the Clinton Global Initiative America, Clinton asked Geithner, “What are your career plans?”

“I live for this work. It’s the only thing I’ve ever done. I believe in it,” the Treasury secretary said. Looking wistful, Geithner in a low tone described the distance from his family he was about to endure and the long commute ahead. “And I’m going to be doing this for the foreseeable future.”

“Good for you,” Clinton said. “That’s good for America!”