Secretary of State Antony J. Blinken has a crisis on his hands. But unlike the foreign coups, hostage-takings and military threats that the nation’s top diplomat routinely faces, this one comes from within the vast bureaucracy he commands — and may be even more difficult to solve.

The problem is a huge backlog of passport applications that is creating summer travel nightmares for Americans who find that getting a new passport or renewing an expired one can take months, forcing them into panicked races against their planned travel date through an often bewildering bureaucratic maze.

Senator Mark Warner, Democrat of Virginia, has called the situation a “crisis.” Senator Rick Scott, Republican of Florida, has said it is “an unacceptable failure.” And Utah’s entire congressional delegation told Mr. Blinken in a letter this spring that their offices were “struggling to handle all incoming emergency requests due to the sheer volume” of pleas from their constituents.

“While running a competent passport application process may not make a panel at Davos, this is an important function of the federal government that directly affects the lives and plans of millions of Americans,” Senator Eric Schmitt, Republican of Missouri, said in a letter to Mr. Blinken, referring to the elite economic forum held annually in the Swiss Alps.

The State Department, which issues and renews passports for American citizens, has said it is still recovering from disruptions caused by the coronavirus pandemic, even as it faces record numbers of applications, driven by surging demand from Americans who let their passports lapse over the past few years.

“We’re throwing everything we can at this, trying to make sure that people have those blue books, that they’re able to travel,” Mr. Blinken said at a news conference last month after a question about the delays punctuated his commentary about matters like Russia, Taiwan and ambassadorial nominations.

“It’s something that comes up repeatedly with members of Congress, with folks that I come across,” Mr. Blinken said.

The State Department says it is receiving 430,000 applications a week, down some from a peak of 560,000 per week in March. The department is on track to issue 25 million passports this year, an increase from last year’s 22 million.

The problem is particularly acute at the moment, with millions of Americans traveling out of the country for August holidays. The State Department’s current estimated processing time of 10 to 13 weeks — or seven to nine weeks for a $60 fee — has blindsided Americans who expect shorter turnarounds. For many who have trips planned this month, it may be too late to procure or renew passports.

“Constituents are reporting that they are placed on hold by passport offices for hours before calls inexplicably drop,” the Utah delegation said in its letter. “Constituents who do get through are being given incorrect information over the phone, such as being told they cannot upgrade to overnight shipping or expedited services.”

The State Department says its goal is to reduce processing times to about six to eight weeks for routine service and two to three weeks for expedited service.

People who need passports urgently can get one faster by making an in-person appointment, rather than relying on the traditional mail-in process. But high demand has made those appointments hard to secure — and some people have been dismayed to learn that timely appointments are available only at sites far from home. (The State Department also offers “life-or-death emergency” appointments.)

In a statement, Mr. Warner expressed some sympathy for the State Department, saying it had been hit by a “perfect storm of events” — particularly the fact that millions of Americans did not renew their passports in recent years because of a pandemic travel lull “and now are trying to renew along with everybody whose passport is expiring in 2023.”

Mr. Warner blamed State Department hiring freezes during the Trump administration for creating staffing shortfalls and said he was pressing for the department to hire more personnel, particularly information technology specialists.

He advised people facing travel emergencies to reach out to “your senator or member of Congress, who may be able to help.”

Showing less sympathy over structural problems, some Republicans have accused Mr. Blinken of neglecting the problem.

In a follow-up letter to Mr. Blinken last month, Mr. Schmitt argued that the State Department’s “misplaced” priorities had led to millions spent on diversity, equity and inclusion programs, efforts to combat wildlife trafficking in China, and $120,000 on “gastrodiplomacy,” all of which he called “expenditures that could be spent on hiring more passport agents.”

Recognizing the problem, Mr. Blinken paid a late June visit to his department’s largest passport processing office, in New York City.

“I saw the operation firsthand and saw people working overtime, double time, triple time to get blue books into people’s hands,” he said. “We’re trying to do our best at getting people their passports.”