WASHINGTON—Both the Justice Department and a group of state attorneys general are likely to file antitrust lawsuits against Alphabet Inc.’s Google—and are well into planning for litigation, according to people familiar with the matter.

The Justice Department is moving toward bringing a case as soon as this summer, some of the people said. At least some state attorneys general—led by Texas Attorney General Ken Paxton, a Republican—are likely to file a case, probably in the fall, people familiar with the matter said.

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