The U.S. Food and Drug Administration is nearing a decision to authorize Covid-19 booster shots for certain people with weak immune systems, people familiar with the matter said, a shift in the American vaccination strategy as the Delta variant drives up cases of infection.

The agency could make a decision as early as Thursday and probably by Friday, one of the people said.

The agency is closing in on the booster-shot authorization amid evidence that vaccines are less effective protecting immunocompromised people from Covid-19 than they are in protecting the general population.

Some public health experts and vaccine makers have also recommended boosters as necessary for broader swaths of the population to keep strong protection against newer variants of the coronavirus, and because some preliminary research indicates vaccines may lose effectiveness over time.

The Biden administration is expected to lay out a plan for boosters by early September.

Some countries, including Israel, have authorized booster shots for immunocompromised patients and older adults.

Last month, an expert panel advising the U.S. Centers for Disease Control and Prevention signaled preliminary support for giving extra doses to immunocompromised people. The panel is scheduled to meet Friday to discuss the matter further.

NBC News, CNN and the New York Times earlier reported that the FDA was expected to authorize the booster shots for immunocompromised people soon.

Three Covid-19 vaccines are authorized for use in the U.S. Shots from Pfizer Inc. and its partner BioNTech SE and from Moderna Inc. require two doses for full protection, while the vaccine from Johnson & Johnson involves a single dose.

Pfizer has said it plans to ask U.S. regulators to authorize booster shots of its two-dose vaccine this month while Moderna has said it plans to do so next month.

Johnson & Johnson is expected to release data about the efficacy of two doses of its single shot vaccine later this month.

The Delta variant, which was first identified in India, has spread rapidly throughout the U.S. in recent weeks. Health authorities blame it for increasing cases—and hospitalizations—across the U.S., especially among people who haven’t gotten vaccinated.

About 2.7% of American adults are immunocompromised, which includes people who have been given transplants, live with HIV or take cancer treatments and other drugs that suppress their immune system.

Their weakened immune systems make the people less responsive to vaccines, which require stimulation of the immune system to provide protection.

Research suggests immunocompromised people are more likely to get severely ill from Covid-19, are more likely to transmit it to household contacts and more likely to have breakthrough infections.

Transplant recipients, who often take a cocktail of powerful immunosuppressant drugs, are among the least likely to develop antibodies from the vaccines, according to doctors and several new studies.

The FDA, in reviewing the matter, has taken note of recent data from Moderna showing a third shot of its vaccine helped increase antibody levels in recipients, according to one of the people familiar with the deliberations.

The agency has been working as quickly as possible on the authorization, the person said, adding that boosters may be necessary for far broader swaths of the U.S. population in fall or winter.

In four studies, between one-third and one-half of the subjects who had no antibody response after the first two doses of a messenger RNA Covid-19 vaccine ended up developing a response after the third, CDC staff said last month at last month’s meeting of the CDC’s vaccine advisory panel.

“The data are pretty solid on this…It’s clear that a third shot really gives them a decent response because they aren’t responding well to two,” Dr. Arnold Monto, a professor of epidemiology at the University of Michigan and acting chair of an expert panel that advises the FDA about vaccines said in an interview.

About half of the country, or 167.1 million people, have been fully vaccinated, according to data from the CDC.

Write to Stephanie Armour at stephanie.armour@wsj.com, Felicia Schwartz at felicia.schwartz@wsj.com and Jonathan D. Rockoff at Jonathan.Rockoff@wsj.com