At 57, Mr. Lew may be the most unassuming power broker in Washington. He is deeply religious (an Orthodox Jew, he leaves work each Friday before sundown) and is so strait-laced that his colleagues feel compelled to apologize when they curse in front of him. He brings his own lunch (a cheese sandwich and an apple) and eats at his desk.

With his owlish glasses and low-key manner, Mr. Lew may come off as just a policy nerd. But he is a fierce negotiator. When defending social safety net programs, particularly those like Medicaid that help the poor, he morphs into a warrior, Republicans say, though he has proved willing to make concessions.

“Jack is tough,” said Jim Dyer, a Republican and a former Capitol Hill aide who negotiated budget issues with Mr. Lew in the 1990s. “He can be argumentative, he’s smart as hell, he’s very political, he is a true liberal, he is loyal to his superiors, and he has a good grasp of budgetary and policy issues.”

“Fighting with him is exhausting,” Mr. Dyer added. “We yelled at each other a lot. We never came to blows. We walked away from the table perhaps happy to be away from each other for a while, but perhaps equally happy that we preserved a modicum of what each side wanted.”

Mr. Lew arrived in Washington in 1973, a skinny, bookish 18-year-old from Queens who got his first taste of Democratic politics at 12 while handing out fliers for Eugene McCarthy’s presidential campaign. Today, as a two-time former budget director (he also held the job under President Bill Clinton), he has an intricate understanding of budget policy.

In 1983, as an aide to Speaker Tip O’Neill when Ronald Reagan was president, Mr. Lew helped put Social Security on a path to solvency with a plan that, to many Democrats’ chagrin, will eventually raise the retirement age to 67. He keeps a gavel from the day the legislation passed, signed by Mr. O’Neill, on a bookshelf in his office.