The cash balance at the Treasury Department’s general account fell below $40 billion last week as lawmakers raced to reach an agreement to increase the nation’s borrowing cap. Mr. Biden on Saturday signed legislation that suspended the $31.4 trillion debt limit until January 2025.

For months, Treasury Secretary Janet L. Yellen had been using accounting maneuvers known as extraordinary measures to delay a default. Those included suspending new investments in retirement funds for postal workers and civil servants.

Restoring those investments is essentially a simple accounting fix, but refilling the government’s cash coffers is more complicated. The Treasury Department said on Wednesday that it hoped to borrow enough to rebuild its cash account to $425 billion by the end of June. It will need to borrow much more than that to account for planned spending, analysts said.

“The supply floodgates are now open,” said Mark Cabana, an interest rate strategist at Bank of America.

A Treasury Department spokesman said that when making decisions on issuing debt, the department carefully considered investor demand and market capacity. In April, Treasury officials started surveying key market players about how much they thought the market could absorb after the debt-limit standoff was resolved. The Federal Reserve Bank of New York this month asked large banks for their estimates of what they expected to happen to bank reserves and borrowing from certain Fed facilities in the next months.

The spokesman added that the department had managed similar situations before. Notably, after a bout of debt-limit wrangling in 2019, the Treasury Department rebuilt its cash pile over the summer, contributing to factors that drained reserves from the banking system and upended the market’s plumbing, prompting the Fed to intervene to stave off a worse crisis.

One of the things the Fed did was establish a program for repurchase agreements, a form of financing with Treasury debt posted as collateral. That backstop could provide a safety net to banks short on cash from lending to the government, though its use was widely seen in the industry as a last resort.