The Trump administration is considering moving responsibility for overseeing more than $1 trillion in student debt from the Education Department to the Treasury Department, a switch that would radically change the system that helps 43 million students finance higher education.

The potential change surfaced in a scathing resignation memo sent late Tuesday night by James Runcie, the head of the Education Department’s federal student aid program. Mr. Runcie, an Obama-era holdover, was appointed in 2011 and reappointed in 2015. He cut short his term, which was slated to run until 2020, after clashing with the Trump administration and Betsy DeVos, the education secretary, over this proposal and other issues.

Elizabeth Hill, a spokeswoman for the Education Department, declined to comment on his departure or on talks with Treasury.

“The secretary is looking forward to identifying a qualified candidate to lead and restore trust in F.S.A.,” Ms. Hill said, referring to federal student aid.

A shift in handling federal student aid is being weighed as the Trump administration and Ms. DeVos consider overhauling the Department of Education. Mr. Trump’s proposed budget for 2018 slashes funding for the department by nearly 50 percent. Moving one of its core functions to Treasury would significantly diminish the agency’s power. It could also alter the mission of the student loan program.

“The reason the federal student aid programs live within the Education Department is because that’s the agency that has as its goal increasing educational opportunities within the United States,” said David Bergeron, who left the Education Department in 2013 after 35 years. “That is not the Treasury Department’s goal. Its job is to pay for the business of the government.”

Scrapping or shrinking the Education Department has long been a popular Republican goal, dating from the Reagan administration. President Trump embraced the idea, saying in his book “Crippled America” that the department should either be eliminated or have “its power and reach” cut. In February, a House Republican introduced a bill to terminate the agency.

In his resignation memo, a copy of which was obtained by The New York Times, Mr. Runcie said that senior members of his department had met that day with Treasury officials and discussed “holding numerous meetings and retreats” to outline a process for “transferring all or a portion” of the student aid office’s functions to the Treasury Department.

“This is just another example of a project that may provide some value but will certainly divert critical resources and increase operational risk in an increasingly challenging environment,” Mr. Runcie wrote.

Moving the federal student aid unit probably would require congressional action. But even in a fractured Congress, it could win bipartisan support.

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The federal student aid office has been a lightning rod for criticism over the effectiveness and expense of its debt collection programs. Several government audits took issue with the department’s handling of its student aid programs. In 2015, for example, the Government Accountability Office faulted the agency for not doing enough to make students aware of all their repayment options. The Consumer Financial Protection Bureau has also pressed for changes in how the department manages its loan servicers.

The Education Department backs and originates $1.4 trillion in student loans. Since 2010, the government has directly funded the loans, cutting out the private lenders that previously doled out government-backed aid. But the agency outsources the work of collecting payments on the loans, and the companies it works with have a troubled record.

During the Obama administration, the idea of shifting responsibility for the student loan program to the Treasury Department had some supporters. As the number and dollar amount of student loans grew, the Education Department found itself managing more than a trillion dollars in assets, a portfolio bigger than most banks.

“The Education Department is a policy shop with a trillion-dollar bank on the side,” said Rohit Chopra, a former student loan ombudsman at the Consumer Financial Protection Bureau who also briefly worked for the Education Department.

For students, the move under consideration could simplify the convoluted process of applying for federal student aid and repaying loans. A growing number of borrowers are using income-based repayment plans, which require students to submit information on their earnings. Putting federal student aid in the same department as the Internal Revenue Service could make that easier. (A tool intended to help students automatically import their tax information has been disabled for months because of a security problem.)

“I think it’s a good idea,” said James Kvaal, a former deputy under secretary of education in the Obama administration. “Because the Education Department and the I.R.S. are separated, we’ve built these clunky systems that get in the way of achieving the goals of the income-based program. Linking the two would be much easier for students, and have stronger integrity for taxpayers.”

But critics, including a high level official from Mr. Obama’s Treasury Department, warned that the move could hurt students.

“Moving the agency that is supposed to provide stewardship for student loan borrowers to an agency that is working on a shoestring with a skeletal crew strikes me as a recipe for a policy disaster,” said Sarah Bloom Raskin, who was the deputy Treasury Secretary under President Obama.

Others worry about how students would fare under the Treasury Department.

The Treasury Department recently conducted a pilot project in which its employees tried to collect on defaulted loans, a job the Education Department contracts out to private companies.

The experiment, which began in mid-2015, did not end well.

The Treasury Department hoped to increase collection rates and help borrowers better understand their repayment options. It failed on both goals. A control group of private collectors recovered more money and got more borrowers out of default.

For now, even without the shift, some at the federal student aid office are rattled, according to one person who requested anonymity because he was not authorized to speak publicly. After Mr. Runcie resigned, at least one employee was in tears, the person said.

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