ISTANBUL—After an early drive to inoculate millions with a Chinese-made Covid-19 vaccine, Turkey started offering Thursday a booster shot to some people amid concerns over the spread of the highly transmissible Delta strain.

Turkey’s health minister, Fahrettin Koca, said it would offer a third vaccine dose to healthcare workers and people over the age of 50. That third dose isn’t dependent on which vaccine was used for the previous two. Turkey is currently using both the vaccine made by Pfizer Inc. and BioNTech SE and another made by the Chinese company Sinovac Biotech Ltd.

“The choice is yours,” said Mr. Koca.

In response to questions from The Wall Street Journal, the Turkish Ministry of Health said that it is offering the booster shot because some healthcare workers and older people received their initial doses six months ago. At the time, Turkey was only offering the Sinovac vaccine.

Turkey isn’t the first country to offer booster shots to people who have received a Chinese-made vaccine. Bahrain has been giving Pfizer-BioNTech shots to people who received two doses of a vaccine made by Chinese drugmaker Sinopharm. The United Arab Emirates is giving third doses of the Sinopharm vaccine as doctors say the Chinese-made shots in some cases haven’t generated enough protective antibodies.

Turkey launched its vaccine campaign in January using the Sinovac vaccine, inoculating millions of people before supplies from China slowed. The country also started vaccinating people with the Pfizer-BioNTech vaccine in March.

In recent weeks, Turkey accelerated its campaign after the country reached agreements to import 120 million doses of the Pfizer-BioNTech vaccine by September, according to the health ministry. The country has been vaccinating an average of 733,238 people a day over the past week, one of the highest daily rates in the world.

In June, Turkey’s President Recep Tayyip Erdogan said he received three shots but didn’t specify which vaccines he took. Mr. Erdogan took his first dose publicly in January at a time when only the Sinovac shot was available, but later said that he had taken the Pfizer-BioNTech vaccine.

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Mr. Erdogan’s office didn’t immediately respond to a request for comment on his choice of vaccines.

The spread of the extra-contagious Delta variant first detected in India has added urgency to vaccination efforts around the world, particularly as the virus spreads among unvaccinated people and those who have received only one dose.

Studies have shown that receiving two doses of some vaccines—including the one made by Pfizer-BioNTech—offers significant protection from the Delta variant. No large-scale data have been released on the effectiveness of Sinovac’s vaccine against the Delta variant.

In Israel, about half of adults infected in an outbreak of the Delta variant were fully inoculated with the Pfizer vaccine. These so-called breakthrough cases are broadly expected because the vaccine—while not 100% effective in preventing infections—is a potent way of suppressing the severity of the illness and reducing hospitalizations, according to public health experts.

Turkish officials have cited the spread of the Delta variant, even in some countries with high rates of vaccination, as a reason to speed their vaccination drive. At a separate news conference on June 23, Mr. Koca said that “not a serious amount” of Delta cases had been detected in Turkey but the government was concerned the number could rise.

Write to Jared Malsin at jared.malsin@wsj.com