Grassley said the vote had been pushed back for a week because of Feinstein's objections.
The legislation was introduced by Republicans Thom Tillis of North Carolina and Lindsey Graham of SC and Democrats Chris Coons of DE and Cory Booker of New Jersey - who are all members of the Judiciary panel. It would also give any special counsel a 10-day window to seek expedited judicial review of a firing.
Recent reports indicated that Trump has been especially irate with an FBI raid of his personal lawyer's office, the findings which could eventually be turned over to Mueller for his investigation into Russian meddling in the 2016 election.
It would write into law the existing Justice Department regulations that say a special counsel can only be fired for good cause and by a senior Justice Department official. Feinstein stated on Wednesday.
"Why don't I just fire Mueller?" "We'll see what happens", he said, speaking to members of the media on Monday. "It's an attack on our country in a true sense".
If the legislation makes it through Congress and earns Trump's signature - a big political "if" - it could face a skeptical Supreme Court. (Attorney General Jeff Sessions recused himself from the investigation related to the 2016 campaign.) If Trump were to sack Rosenstein, the deputy's successor would take control over the special counsel and scope of his investigation. Over the past few months, the special counsel's office has issued indictments for several Trump campaign associates, as well as White House national security adviser Michael Flynn, for various charges, including lying to the Federal Bureau of Investigation about their ties with Russian officials.
Agents are also believed to be looking into how Cohen obtained the money and whether he committed financial fraud in the process. Her statement came after Trump suggested, talking to reporters that he could fire the Special Counsel, since "it's a disgrace what's going on".
"I (we) are doing things that nobody thought possible, despite the never ending and corrupt Russia Investigation, which takes tremendous time and focus", Trump tweeted on Wednesday. In some cases, they have explained to Mr. Trump how anything that causes him to lose support from congressional Republicans could further imperil his presidency.
The raid carried out by the Federal Bureau of Investigation on the NY office of Trump's lawyer identified as Michael Cohen was described by the United States President as "disgraceful".
Even if Trump is tempted to fire Rosenstein, his tweet Thursday indicates he'll give Bannon zero credit for the idea, as sources told the Post there is still bad blood. He has repeatedly backed the special counsel, and signed off on the raid of Cohen's properties, according to a person familiar with the matter.