Lawmakers will soon leave Washington for a recess, meaning no vote would be imminent - if the resolution made its way through procedural hurdles to a vote.
House Republicans have been ramping up their attacks on the deputy attorney general in recent weeks, accusing him of withholding documents and being insufficiently transparent in his handling of the probe led by special counsel Robert Mueller III.
Democrats have argued the focus on Rosenstein is an attempt to broadly undermine the Mueller investigation. The Republicans in question have no quarrel with Trump on this matter, and if he wanted to, he could simply fire Rosenstein - which would be much easier than impeaching him.
Jim Jordan is shown during a House committe hearing on July 12. Elaborating further, the congressman said he hopes to avoid impeachment.
Attorney General Jeff Sessions and others hailed Rosenstein, and he appeared to remain safe in his job.
The Justice Department said it had no comment. The lawmakers alleged bias at the Federal Bureau of Investigation and suggested the department has conspired against Trump - but many could draw the line at impeachment.
The resolution notes that Rosenstein was one of several department officials who approved what some Republicans say was improper surveillance of Trump campaign adviser Carter Page. The move comes after months of criticism over how the investigation is being carried out from US President Donald Trump and his allies. "It's an absurd idea", Schiff, D-Calif., said. "I also don't think it proves anything about collusion or anything like that".
"We want to make sure that we get compliance".
Rosenstein is the top DOJ official overseeing special counsel Robert Mueller's Russian Federation investigation.
Oversight Chairman Trey Gowdy, who says he's opposed to trying to impeach Rosenstein, said that DOJ has "provided a lot of documents", but it's still not full compliance.
"The stonewalling over this a year ago has been just as bad [as] or worse than under the Obama administration", Meadows said in a statement.
He added the impeachment process could disrupt the legislative process and affect other Congressional objectives.