WASHINGTON—President Trump on Friday chose as his next attorney general William Barr, who served that role during the first Bush administration, settling on a veteran Republican who has long held sweeping views of presidential power to oversee the special counsel’s Russia investigation as it continues to roil the administration.

Mr. Barr, if confirmed by the Senate, would succeed acting attorney general Matthew Whitaker, who in earlier writings and television appearances had criticized the investigation into Russian election...