Washington — When members of Congress proposed legislation that would deny federal matching funds to states that cut their higher-education budgets, lobbyists for state governments went ballistic.

The National Governors Association, in a letter sent to Capitol Hill in December, called the provision, which lawmakers had added to a bill to reauthorize the Higher Education Act, an “unprecedented overreach of federal authority that is unworkable for state budgets.” And the National Conference of State Legislatures, in a series of press releases, said the plan would “set a dangerous precedent of federal intrusion into state budgeting decisions.”

It seems at least one lawmaker was listening.

When the U.S. House of Representatives takes up the reauthorization bill on Thursday, Rep. Rob Bishop, a Republican of Utah, will offer an amendment that would strike the “state maintenance of effort” language.

The amendment is one of 61 that the committee will consider to the bill, which would reauthorize, or renew, most higher-education programs for five years.

But if state governments may be able to breathe a bit easier, colleges may not.

Among the amendments scheduled to be offered is one by Rep. Peter Welch, a Democrat of Vermont, that would require colleges to spend at least 5 percent of their endowments on student aid. The plan, which originated in the Senate Finance Committee, is fiercely opposed by colleges, which say the government should not be meddling with their endowments.

Mr. Welch also plans to offer an amendment that would require colleges to report annually on how much of their endowment is paid out each year, what the money is spent on, and how much of the payout is used to defray the cost of college. —Kelly Field