WASHINGTON—Former Exxon Mobil Corp. chief executive Rex Tillersoncleared an important hurdle Sunday on his path to become secretary of state, bolstering Senate Republicans’ expectations they will be able to confirm most, if not all, of President Donald Trump’s cabinet nominations.

GOP leaders are still unclear when the names will be brought before the Senate for a vote, which means Mr. Trump begins his presidency with a much thinner cabinet than his predecessor.

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