The Republican chairman of the House Intelligence Committee ignited a new battle over President Donald Trump’s claims that he was spied on by the Obama administration, saying that U.S. intelligence agencies intercepted information about people involved in the Trump transition team.

Rep. Devin Nunes (R., Calif.), who is leading a congressional investigation into alleged Russian interference in U.S. elections, shared his information with the media and Mr. Trump before he gave it to other members of his committee. The move drew...