An anti-inflammatory drug used to treat severe cases of Covid-19 is in short supply globally as infections and hospitalizations surge due to the highly transmissible Delta variant.

Genentech, a U.S. subsidiary of Roche Holding AG , said this week that two dosage strengths of the drug, called tocilizumab, were already out of stock across the U.S., with a lower strength form likely to run out by the end of the week. The Food and Drug Administration added tocilizumab, which is sold in the U.S. under the brand name Actemra, to its drug shortages list on Tuesday.

Roche said its capacity to raise production to meet rising demand has been constrained by raw-material shortages and the labor-intensive process of making the drug, a so-called biologic medicine made from living cells. The Swiss drugmaker said in a statement that it expects global shortages to continue over the weeks and months ahead.

Tocilizumab is used in conjunction with steroids to dampen a haywire immune response that occurs in some severely ill Covid-19 patients. The drug, which was originally developed to treat inflammatory diseases such as rheumatoid arthritis, received emergency-use authorization from the FDA in June.

The shortage relates to a formulation of tocilizumab that is delivered by intravenous drip, the company said. An alternative form of the drug that can be delivered by injection is still available, but that version isn’t authorized for use for Covid-19.

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Covid-19 cases and hospitalizations are growing steadily as the highly infectious Delta variant spreads quickly across the U.S., mainly among the unvaccinated. U.S. demand for tocilizumab has more than quintupled in the past two weeks, driven by the surge in Delta-related cases, Roche said.

The shortage comes despite efforts by Genentech and Roche to increase production of the drug, which was identified early in the pandemic as a potential treatment for certain severe cases of Covid-19. Roche said its manufacturing plants are running near capacity.

Genentech said it was working to give priority to areas with the highest need and had directed its remaining stock to hospitals and clinics in Covid-19 hot spots.

The company expects to replenish stocks by the end of August. However, it said that if cases continue to rise at their current pace, there would likely be further shortages in coming months.

Studies on the effectiveness of tocilizumab in preventing death resulting from Covid-19 have produced mixed results. In July, the World Health Organization recommended it for use in severe cases after concluding that it and a similar drug called sarilumab reduced the risk of death by 13% in the sickest patients when taken in combination with steroids.

Adarsh Bhimraj, an infectious-disease specialist at the Cleveland Clinic, said the shortage is affecting hospitals around the country, especially in southern states such as Texas and Mississippi that are struggling with a surge of new infections and hospitalizations.

For Covid-19 patients who don’t get better with steroids and wind up on ventilators, tocilizumab is the only option for dealing with the hyperinflammation caused by the virus, he said.

“A lot of places are running out of tocilizumab,” he said. “We are all discussing what we should do.”

Write to Denise Roland at Denise.Roland@wsj.com and Joseph Walker at joseph.walker@wsj.com