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.