WASHINGTON—The first defendants in a criminal investigation of Russia’s meddling in the 2016 presidential campaign could be taken into custody as soon as Monday, people familiar with the matter said, though the nature and target of the charges couldn’t be determined over the weekend.

Friday, prosecutors led by Robert Mueller, the special counsel in the Russia matter, obtained a grand jury indictment against at least one person. Mr. Mueller’s team obtained the charges under seal, the people familiar said.

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