Mr. Mueller has expressed interest in speaking with other administration officials, including members of the communications team. But Mr. Trump’s allies are particularly concerned about Mr. Mueller’s interest in talking to Mr. Priebus, a former chairman of the Republican National Committee who worked closely with Mr. Trump during the presidential campaign. Mr. Trump’s confidants at the White House say Mr. Trump was never fully convinced that Mr. Priebus would be loyal to him.

Shortly after the November election, Mr. Priebus was made chief of staff, and he was involved in the major decisions the president made during the transition and in the first six months of the administration. Mr. Priebus made a point of being in most meetings and tried to be aware of what the president was doing. Mr. Trump fired him last month.