"You cannot have the guy who is, in effect, running the Justice Department, in front of subordinates, talk about recording the president, even if it is done in a sarcastic way," Rep. Jim Jordan, R-Ohio, a leading member of the House Freedom Caucus, said Tuesday evening.
The effort comes at a fraught moment for Rosenstein, who will meet with Trump on Thursday in Washington.
On Monday, he traveled to the White House with the expectation of being fired after an explosive report by The New York Times that he discussed recording Trump and the 25th amendment in 2017, after Trump fired then-FBI Director James Comey.
"I never pursued or authorized recording the President and any suggestion that I have ever advocated for the removal of the President is absolutely false," Rosenstein said in a statement responding to the report.
Jordan, who is leading the effort with House Freedom Caucus Chairman Mark Meadows, R-N.C., a Trump confidant, has been pushing Republican leaders to ask Rosenstein to appear on Capitol Hill under oath for several days.
"I find no compelling reason why we shouldn't have Rod Rosenstein come before Congress and put up his right hand, and say what he did or didn't say," Meadows said Tuesday. "I get a sense [from leadership] that there's not an appetite for doing it."
The Republicans are contemplating forcing a vote on impeaching Rosenstein if they don't get their way, using a privileged impeachment resolution, which they first considered using earlier this year after a standoff with the Justice Department over sensitive documents related to the Russia investigation.
"All options have been on the table. Certainly any option in our tool bag is certainly an option that we can still deploy," Meadows said, adding that no decision had been made.
It's unlikely the House would vote to table the impeachment resolution, if introduced, before leaving Washington for the month on Friday.
House Judiciary Committee Chairman Bob Goodlatte, R-Va., did not answer questions about Rosenstein's appearing before Congress.
"We've made no decisions, but I understand that he's meeting with the president on Thursday," he said.
Goodlatte said he notified committee Democrats of his intention to subpoena the Justice Department for the memos of former FBI Deputy Director Andrew McCabe as soon as Thursday. The Justice Department had previously refused to turn over the records to Congress.
The notes detail McCabe's conversation with the deputy attorney general and Rosenstein's suggestion that McCabe wear a wire to record the president, sources familiar with the memos have told ABC News.
"Those are documents that would tell us something about that meeting without having to deal with the testimony of anybody," Goodlatte said.