WASHINGTON—Christopher Wray, President Donald Trump’s nominee to be the next FBI director, cleared a key hurdle in his confirmation process on Thursday when the Senate Judiciary Committee voted in a show of unity to advance his nomination to the full Senate.

The unanimous vote by the 11 Republicans and nine Democrats on the panel paves the way for Mr. Wray to be confirmed and take office as the next leader of the Federal Bureau of Investigation before the Senate adjourns in mid-August for a brief summer recess. Mr. Wray, whom Mr. Trump nominated last month after firing James Comey, has drawn bipartisan praise on Capitol Hill and has faced no serious obstacles from his confirmation in the Senate.

He inherits leadership of the FBI from Mr. Comey, who was fired by the president in the middle of an investigation into Russian activity during the 2016 presidential election and whether Mr. Trump or his associates had contact with Moscow. Mr. Comey’s firing alarmed many senators in both parties.

An attorney by training, Mr. Wray has worked in private practice and the U.S. government. He has spent much of the last decade representing high-profile clients including New Jersey Gov. Chris Christie and Credit Suisse Group AG . Before that, he was a federal prosecutor in Georgia and served in the Justice Department leadership after the Sept. 11, 2001, terrorist attacks. In 2003, he was tapped to run the criminal division of the Justice Department.

Mr. Wray’s testimony before the Senate Judiciary Committee last week earned him bipartisan praise.