The official said that Priebus' request came after the FBI told the White House it believed a New York Times report last week describing those contacts was not accurate.
As of yesterday, the FBI had not stated that position publicly and there was no indication it planned to.
The New York Times reported that US agencies had intercepted phone calls last year between Russian intelligence officials and members of Trump's 2016 campaign team.
"The White House is simply not permitted to pressure the FBI to make public statements about a pending investigation of the president and his advisers," said Michigan Rep John Conyers, the top Democrat on the House Judiciary Committee.
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A 2009 memo from then-Attorney General Eric Holder said the Justice Department is to advise the White House on pending criminal or civil investigations "only when it is important for the performance of the president's duties and appropriate from a law enforcement perspective."
The White House official would not comment when asked if the administration was concerned about the appropriateness of Priebus' communications with McCabe. The official was not authorised to disclose the matter publicly and insisted on anonymity.
The FBI would not say whether it had contacted the White House about the veracity of the Times report. CNN first reported that Priebus had asked the FBI to weigh in on the matter.
Trump has been shadowed by questions about potential ties to Russia since winning the election. US intelligence agencies have also concluded that Russia meddled in the campaign to help Trump defeat Democrat Hillary Clinton.
Still, Trump and his advisers have denied contacts with Russian officials during the election. Last week, Trump said "nobody that I know of" spoke with Russian intelligence agents during the campaign.