Gilead Sciences Inc. will stop considering individual emergency requests for its experimental coronavirus drug, except in certain circumstances, as it grapples with overwhelming demand, the company said Saturday.

The move is part of the drugmaker’s overhaul of the process for doctors requesting emergency use of the experimental drug remdesivir outside of clinical trials. Gilead said it is making the changes because it can’t handle all the requests it is getting, and hoping the revamping will eventually speed up the process and expand access to more patients.

Gilead and other researchers are conducting multiple clinical trials of remdesivir in patients hospitalized by the virus. Initial study data will be available in the coming weeks, Gilead Chief Executive Daniel O’Day wrote in an open letter posted on the company’s website.

Not all patients can get into the studies, either because they don’t meet the clinical eligibility criteria or aren’t being treated at a hospital participating in the trials. For severely ill patients who can’t get into a study, doctors can make a “compassionate use” request for unapproved drugs, which have to be evaluated on a case-by-case basis.

“This works well when there is only a limited number of requests—as is normally the case—but the system cannot support and process the overwhelming number of applications we have seen with Covid-19,” Mr. O’Day said in the letter, referring to the disease caused by the new coronavirus.