As law schools slowly recover from the worst legal job market in decades, they are fighting with their main overseer over what exactly counts as a job.

Some schools have been paying to place graduates at nonprofit and government organizations until they find permanent jobs. The positions help improve schools’ standings in the job-placement metrics that figure heavily in a widely watched national ranking, but they may not be as meaningful to students hoping to put their degrees to work.

Now, the American Bar Association’s accrediting arm is further tightening the rules on how such jobs are counted. After telling schools this spring they will have to report the fellowships separately from positions found on the open market, it is going further by proposing that these jobs not fully count unless they are expected to last a year and pay an annual salary of at least $40,000. Otherwise, they get tallied in a short-term category.

ABA accreditors plan to meet Friday in Chicago to weigh the proposal.

The dispute highlights the growing tension between schools, their students and accreditation authorities amid a sharp decline in legal employment. The employment rate is down to 86.7% for class of 2014 graduates, compared with almost 92% for the class of 2007, according to the National Association for Law Placement, an independent jobs-data group. Students are reluctant to spend the $100,000 or more often needed to cover a law degree without some assurance they will come out on the other end with a job.

Meanwhile, declining enrollment is forcing schools to compete ever more fiercely with each other for students.

School-funded positions accounted for 3.2% of all full-time, long-term legal jobs for the class of 2014, the ABA said, up from 2.9% the previous year. The vast majority of those positions come from 25 law schools.

Emory University School of Law funded 52 of those jobs for the class of 2014, making up 23% of the 224 full-time, long-term legal positions its graduates found.

The school’s dean, Robert Schapiro, said in a comment letter to the ABA that the new proposals appear to be “driven by a desire” to label the so-called bridge-to-practice positions as “bad” jobs versus “good” jobs. The accrediting body, the letter says, should not “place subjective value judgments on differing employment opportunities.”

Christopher Pietruszkiewicz, chair of the ABA committee that came up with the proposed changes, said the group’s task is to provide current and prospective students with accurate information and not to assess the value of these jobs.

Among the class of 2014 graduates, the ABA reports that nearly 60% found full-time, long-term employment requiring a law license, with another 11.2% finding full-time, long-term jobs that give preference to those with a legal degree.

Mr. Pietruszkiewicz, the dean of Stetson University College of Law in Florida, said the committee re-evaluated the rules because it believed the way some schools characterized law school-funded jobs was potentially misleading to applicants.

By setting the $40,000 salary minimum, the committee is hoping to weed out positions that pay so little students wouldn’t want them long-term, he said. The majority of school-funded positions paid stipends that amount to between $12,000 and $28,860 a year, according to NALP.

Patrick Boyle, 25 years old, tapped into funding from USC Gould School of Law in Los Angeles after graduating last year to take a position with the Arizona attorney general’s office in Phoenix. Earlier this month, he officially joined the staff in the office’s child and family protection division.

The $2,000-a-month USC stipend, while tough to live on, “let me get in the door” at a place that wouldn’t have hired him otherwise, he said. Government organizations “don’t really want to subsidize training if they don’t have to.”

Law schools are motivated in part by how school-funded jobs will affect their outcome in the influential U.S. News & World Report rankings. That in turn affects their ability to attract students from a shrinking pool. The number of law school applicants dropped to 55,700 in 2014, according to the Law School Admission Council—down 36.6% from four years earlier.

“One of the reasons law schools do this is to help their numbers,” said Martin Katz, dean of University of Denver Sturm College of Law. The jobs also help students gain work experience and give budget-strapped public interest organizations a needed employee. But if the jobs no longer count as full-time employment, as the new rules propose, “schools are going to be less inclined to do this,” Mr. Katz said.

Denver Law funded 32 positions for the class of 2014, though only two were full-time, long-term legal jobs.

While law schools don’t know the ranking’s exact methodology, Robert Morse, the chief data strategist at U.S. News, said he no longer gives school-funded positions the same weight as jobs found on the open market.

The tension over how to define school-funded jobs highlights the tremendous stress on legal education right now, said James Leipold, executive director at NALP. “It’s the survival of the fittest, it’s a bloody arena,” he said. “Law schools are battling for a diminishing number of students.”