When Janet L. Yellen was Federal Reserve chair in 2014, she faced a grilling from Republicans about whether the federal government had a plan if the nation’s borrowing limit was breached and measures to keep paying the country’s bills were exhausted.

Ms. Yellen, appearing at a congressional hearing, outlined a dire scenario in which financial institutions might try to make payments that they could not cover, because the Treasury Department was out of money, leading to a cascade of bounced checks. She pushed back against the notion held by some Republicans that an economic meltdown could be averted, warning that there was no secret contingency plan.

“To the best of my knowledge, there is no written-down plan,” Ms. Yellen said at the time, adding that it was beyond her remit at the Fed. “That’s a matter that is entirely up to the Treasury.”

Fending off such a calamity is now squarely the responsibility of Ms. Yellen, who is confronting the biggest test she has faced in her eight months as President Biden’s Treasury secretary. Mr. Biden chose Ms. Yellen to help steer the economy out of the pandemic downturn. But in the face of congressional dysfunction, she has been thrust into a political role, trying to convince reticent Republican lawmakers that their refusal to lift the debt cap — which limits the government’s ability to borrow money — could lead to a financial collapse.

It is not a comfortable spot for Ms. Yellen, an economist by training who is now trying to navigate the rough political waters that she tends to avoid by countering legislative gamesmanship with economic logic.

Over the past month, Ms. Yellen has reached out to Democrats and top Republican leaders, including Senator Mitch McConnell of Kentucky, the minority leader, and Representative Kevin Brady of Texas, the top Republican on the Ways and Means Committee. She has used those calls to convey the economic risks, warning that the Treasury’s ability to stave off default is limited and that failure to lift or suspend the debt cap by sometime next month would be “catastrophic.”

Ms. Yellen has reminded Republicans in the calls that they have been willing to join Democrats in lifting the debt ceiling in the past, and that raising the cap allows the U.S. to pay its existing bills and does not authorize new spending.

Thus far, Republicans seem unmoved by Ms. Yellen’s overtures.