When European Union antitrust chiefMargrethe Vestager fined Alphabet Inc.’s Google €2.4 billion last June for abusing its dominance to clobber rivals, Josh Hawley in Missouri took notice.

Mr. Hawley, the state’s attorney general, launched his own antitrust investigation into the same allegations in November and has demanded a copy of all evidence Google has given the EU. Other state attorneys general are talking to Missouri about sharing information, according to people familiar with the discussions. “We are all watching the...