“The fallacy of this formula is that the more you spend, the more you get. Where’s the incentive to have linoleum floors instead of marble?” said Wick Sloane, an English teacher at Bunker Hill Community College who co­authored a 2006 paper for The Center for College Affordability & Productivity that advocated for open debate on the rates.

Competition is fierce. A cottage industry of consultants, accountants, and attorneys has exploded to help schools maximize the amount of money that can be squeezed from the federal government. Some smaller institutions lacking the resources to fully engage in the tedious negotiations favor a flat rate.

A complete picture of the varied rates and competitive environment is difficult to capture. A government database of universities’ negotiated rates is shielded from the public for “proprietary” reasons, even though most schools post their rates on their websites.

Even the Washington-based Council on Governmental Relations, which represents about 200 research universities in their dealings with federal regulations related to research funding, has stopped publishing the rates in their annual survey, deeming the information too sensitive.

“Our only concern is whether the research funding agencies and the federal government pays its fair share,” said David Kennedy, director of costing policy and studies at the Council on Governmental Relations.

The White House did not specify a number for a flat rate, but some proponents suggested 50 percent, near the national average, as a cap. A cap at that level would cost Harvard about $47 million a year in lost government payments.

The Health and Human Services official said another option being considered was lowering every school’s reimbursement by 3 percentage points, or paying schools 95 percent of their current negotiated rates.

“The flat rate was a nonstarter. It was all about saving the government money, and it wasn’t about helping the universities,” said Tobin Smith, vice president for policy at the Association of American Universities, which spent $108,000 lobbying on an array of government issues last year. The exclusive 113-year-old organization represents 60 of the nation’s leading research universities, including Harvard, MIT, Brandeis, and Boston University.

Dozens of universities, including MIT, submitted letters voicing their opposition to a flat rate.

Universities and the federal government became partners in research after World War II. MIT was one of the biggest beneficiaries. The National Institutes of Health initially set the overhead reimbursement rate at 8 percent and it eventually rose to 20 percent for all universities. In 1965, Congress permitted schools to directly negotiate their rates with the federal government, setting ballooning rates on their course.

Overhead cost rates came under congressional scrutiny in 1992, when federal auditors discovered that dozens of universities, including Harvard and MIT, had inappropriately billed the government for items unrelated to research such as sports tickets, parties, even a university yacht. The universities ended up reimbursing the government a portion of those costs. The rules about how such rates are calculated were subsequently tightened, and some universities’ rates, which once soared into the 90 percent range, were lowered.