The hospitals’ moves come as some health insurers also restrict access to the therapy—unusual pushback against a drug targeting a devastating disease like Alzheimer’s that has few effective treatments.

While some doctors have been eager to start prescribing the newly approved drug, others have criticized the FDA for clearing the drug before studies proved it works. The critics have also expressed concerns about whether the drug’s benefits, which appeared to be modest in studies, are worth the risks of side effects such as brain bleeding that require regular monitoring by physicians.

“Clinical studies failed to demonstrate the effectiveness of Aduhelm…while documenting significant risks, like brain swelling and bleeding,” said a spokeswoman for Blue Cross and Blue Shield of North Carolina, which won’t cover the drug for its commercially insured patients.

Federal officials recently began a monthslong review of whether Medicare will cover Aduhelm and under what circumstances. The vast majority of patients expected to get the drug are on Medicare, the federal insurance program for the elderly and disabled.

The Cleveland Clinic won’t carry the drug in its pharmacy or provide infusions of it to patients following a review of available scientific evidence by a multidisciplinary panel of experts, a hospital spokeswoman said.

Cleveland Clinic doctors, the spokeswoman said, can still prescribe the medicine, but patients will have to receive their infusions at an outside facility.

Aduhelm is given with a monthly infusion, typically at an outpatient medical center. Biogen priced the drug at $56,000 a year, though one health researcher said it will probably cost more for a typical patient.

“Based on the current data regarding its safety and efficacy, we have decided not to carry Aducanumab at this time,” the Cleveland Clinic said. The hospital said it will reconsider the decision when more data are available.

Mount Sinai said it won’t infuse Aduhelm until it sees the findings of a U.S. government investigation into interactions between FDA staff and Biogen during the review process. The investigation, which was requested last week by FDA Acting Commissioner

Janet Woodcock,

has raised concerns about the integrity of the approval process, Mount Sinai said.

“Medical decisions should be based on science and data, so it is disappointing that patients living with Alzheimer’s disease may reportedly not be able to access Aduhelm at some facilities,” a Biogen spokeswoman said. “Biogen continues to stand 100% behind Aduhelm and the clinical data that supported approval.”

Jason Karlawish, an Alzheimer’s specialist at the University of Pennsylvania, said the decision of some health systems to not give Aduhelm suggests an erosion of trust in the FDA’s decision making.

“It’s very disturbing that we’re starting to hear health systems that rely on the FDA sending signals that they don’t trust the FDA,” Dr. Karlawish said in an interview. “I have to trust the system that puts drugs into the pharmacy so that I can prescribe them with confidence.”

The New York Times earlier reported that Cleveland Clinic and Mount Sinai weren’t going to administer Aduhelm to patients.

In June, the FDA approved Aduhelm based on two large but inconclusive studies of its effect in slowing cognitive decline in people with mild Alzheimer’s symptoms.

The agency issued the approval using a regulatory mechanism that allows for drugs to be cleared before they are definitively proven effective, saying the Aduhelm was reasonably likely to provide a benefit by reducing levels of a sticky protein called amyloid from the brain.

The FDA approval was made over the objections of some of its own statisticians and members of an outside committee of experts convened by the agency to provide advice on the drug. Three of the outside advisers resigned from the committee in protest of the FDA’s decision.

The FDA subsequently narrowed its recommendation for who should get the drug, to Alzheimer’s patients in the early stage of the disease.

Dr. Woodcock’s request for an investigation last week prompted Mount Sinai to hold off on treating patients with Aduhelm.

A Mount Sinai spokeswoman said that in addition to completion of the investigation, the hospital is awaiting the writing of “best practices” guidelines for Aduhelm by its own experts, as well as a standard review necessary for adding Aduhelm to its formulary of available drugs.

Sam Gandy, a Mount Sinai professor of Alzheimer’s disease research, said he and his colleagues, who are setting guidelines for Aduhelm, were open to prescribing the drug for certain patients, although only one of Biogen’s two clinical studies appeared to show the drug worked.

Dr. Woodcock’s request for an investigation, however, threw into question whether the drug was appropriately evaluated and approved in the first place, Dr. Gandy said. Over the following days, the group decided they would wait for the investigation’s conclusion before considering infusions.

“I’m willing to allow for the possibility that the FDA may have the latitude to approve [Aduhelm] along this accelerated approval pathway, but not if I can’t trust the integrity of the process,” Dr. Gandy said in an interview. “We still want to do what’s best for our patients, but we simply don’t want to make a decision we can’t sustain.”

Washington neurologist Nancy Isenberg, one of the Providence doctors involved in the decision to not administer Aduhelm, cited the drug’s potential for serious side effects, lack of clear benefit and high cost as reasons behind the policy. Providence, whose system includes 52 hospitals and more than 1,000 outpatient clinics, is also wary of prescribing the drug before investigators return results of an investigation of Aduhelm’s regulatory approval, she said.

“I wish and I hope we can offer safe, effective and affordable treatments” for Alzheimer’s patients, Dr. Isenberg said. “At the same time, this is not that.”

Resistance to Aduhelm from some hospitals and insurers may not affect Biogen, which analysts project could eventually ring up billions of dollars in sales from the drug.

A fair number “of academic medical centers etc. can stay on sidelines and [the] math may still work,” Evercore ISI analyst

Umer Raffat

said in a research note.

Some skeptical doctors, he added, have said they won’t deny Aduhelm to interested patients.

—Melanie Evans contributed to this article.

Write to Joseph Walker at joseph.walker@wsj.com