Ryan made the comments a day after the group of 11 House Republicans sharply escalated the extended clash with the Justice Department by filing articles of impeachment against Rosenstein, who oversees special counsel Robert Mueller's investigation.
Meadows and Jordan passed up an opportunity to force a quick vote on impeachment by opting against pursuing a "privileged" resolution, a procedural tactic that allows members to demand a vote within two legislative days.
The House of Representatives departed on Thursday for a five-week recess after Mr Ryan declined to schedule a vote on the bill. Other documents requested in the subpoena were so sensitive that the Federal Bureau of Investigation had to build a new search tool to gather the materials lawmakers requested, officials explained Wednesday, which accounted for some of the delays.
Republican leaders in the House, including Speaker Paul Ryan, have been clear that they don't wish to impede the Mueller investigation. Usually it's up to the House Judiciary Committee to handle impeachment investigations, but a different or special committee could handle that. He didn't rule out trying for an impeachment vote in the future. Exiting the meeting, Meadows said there was "still great frustration" with the pace that subpoenaed documents had been produced to Congress.
Ryan's tone was far different. "But we've been making tremendous progress to that point", he said.
While Special Counsel Robert Mueller is spearheading the Russian Federation probe, Rosenstein, 53, still oversees the federal investigation in his role as deputy attorney general.
The House Judiciary Committee and House Oversight and Government Reform Committee launched a joint investigation past year into decisions made by the DOJ and FBI.
YouGov Poll: UK voters support a referendum on final Brexit deal
May has said that the United Kingdom can not remain within the Customs Union of the EU, describing it as one of her "red lines". May has ruled out a customs union and Raab said Britain still believed it could reach a deal to avoid a "hard border".
It is unclear whether there would be enough support in the party to pass his resolution if a vote were held.
"Trumps's Congressional coverup crew moved to impeach Rosenstein".
Ryan told reporters Thursday he opposed the impeachment resolution for a number of reasons.
Earlier, Rosenstein's boss, Attorney General Jeff Sessions, expressed confidence in the career civil servant and took a swipe at the lawmakers pushing for his ouster.
One of the subpoenas sought information on surveillance of former Trump campaign adviser Carter Page.
Eleven House Republicans have introduced articles of impeachment against him.
"Technically I could still file it today, I am not doing that", he said. "And we look at all the various tools that we have available, including having the kinds of dialogues we've been having recently in order to get compliance from DoJ", he added. Rod Rosenstein has repeatedly failed to deliver requested documents, ignored subpoenas and ultimately stonewalled Congress.