Mr. Strzok, who was briefly part of Mr. Mueller's team, made national headlines late a year ago when texts sent to his lover, former Federal Bureau of Investigation attorney Lisa Page, were made public.
Lisa Page - whose affair with FBI agent Peter Strzok has led Trump to dub the pair the "FBI lovers" - was due to give a deposition before the House Judiciary and Oversight Committees Wednesday over her role in those investigations.
Strzok and Page were involved in a romantic relationship in 2016, when they were both involved in the investigation of former Secretary of State Hillary Clinton's email server.
Mueller, a Republican, removed Strzok from the inquiry when he learned about the text exchanges and Page has left the Federal Bureau of Investigation.
Judge Andrew Napolitano said that former Federal Bureau of Investigation attorney Lisa Page's plan to defy a subpoena to testify before lawmakers is confusing.
"Instead of responding to our requests to explain the scope of the interview and provide sufficient notice to allow her to prepare, the Committees chose to issue a subpoena", Page's attorney wrote. "Together with her lover, FBI Agent Peter Strzok, she worked on the Rigged Witch Hunt, perhaps the most tainted and corrupt case EVER!" Her attorney said Page hadn't been given enough information about the committees' intended line of questioning.
"We will use all tools at our disposal to obtain her testimony", he added.
He is scheduled to appear again in a public hearing Thursday morning.
"As a result, Lisa is not going to appear for an interview at this time", Jeffress in a statement.
"It appears that Lisa Page has something to hide", he said. But a subpoena to testify before Congress is not optional.
The two exchanged text messages saying "we'll stop" a Trump presidency, a report by the FBI Inspector General found.
Jeffress said in a statement prior to the contempt threat that Page could not appear without first having access to Federal Bureau of Investigation documents that the committees already possessed, and that they had ignored her efforts to find a later date for the deposition.
The Judiciary and Oversight panels have already spent much of the summer holding hearings and interviews critical of the Federal Bureau of Investigation and Justice Department.
Trump torched Page and Strzok on Tuesday while aboard Air Force One headed to Brussels.
Mr. Strzok is scheduled to testify Wednesday before both committees.
"Americans across the country are alarmed at the bias exhibited by top officials at the Justice Department and FBI", Goodlatte said. That includes showing up at a public hearing with Strzok on Thursday, appearing for a closed door deposition Friday or facing more legal trouble.
The committees have focused much of their ire on Strzok, who was interviewed privately by lawmakers on the two committees for 11 hours June 27.
House Republicans have engaged in a war of words and threats against the Justice Department for withholding information about the Trump-Russia investigation.
The Judiciary Committee also held a contentious hearing June 28 with FBI Director Christopher Wray and Deputy Attorney General Rod Rosenstein.