The 84-year-old Feinstein, who is up for reelection in 2018, faces a primary challenge from California's progressive State Senate President Pro Tem Kevin de León.
The senator apologized for not having consulted the committee's chairman, Sen Charles Grassley (R-Iowa), who blasted the maneuver.
Simpson made the claim during a closed-door session with the Senate Judiciary Committee in August, where he was questioned for 10 hours over the dossier and its author, former British spy Christopher Steele.
US President Donald Trump on Wednesday accused Democratic Senator Dianne Feinstein of being "underhanded and a disgrace" for disclosing details of a dossier of allegations about his ties to Russian Federation during the campaign.More news: Here's How The British Royals Hid The Crown Jewels From Nazis
"When they have no collusion and nobody's found any collusion at any level, it seems unlikely that you'd even have an interview", Mr Trump said during a joint news conference with the prime minister of Norway.
The result was the explosive, much-discussed and unverified dossier.
A January 2017 assessment by the Office of the Director of National Intelligence (ODNI) stated that Russian leadership preferred presidential candidate Donald Trump over Hillary Clinton, and that Russian President Vladimir Putin personally ordered an "influence campaign" to harm Clinton's electoral chances and "undermine public faith in the USA democratic process". The file includes salacious details that could have embarrassed Trump and been used as leverage over him.
Cohen, who is named in the dossier, says BuzzFeed and several of its staffers defamed him by publishing the 35-page document.
If that reporting holds up, then the "human source from inside the Trump organization" did not know that he was a human source. Intel officers had briefed both President Obama and President-Elect Trump on the contents and the Federal Bureau of Investigation was already on board.More news: £13m could bring Naby Keita to Liverpool in January
Instead of being the source of the FBI investigations, Simpson claimed that the FBI was already on to the Russians, not least because our Australian allies warned us that the Russians claimed to have dirt on Clinton, which they learned because George Papadopoulos, a Trump advisor, was bragging about it to an Australian diplomat.
"I can't think of a law it violates", said Solomon Wisenberg, former deputy independent counsel for the Whitewater investigation.
This is likely what prompted Feinstein to release the transcripts - which, incidentally, excluded not only a number of important contextual exhibits from Simpson's testimony but also redacted all sorts of relevant information about the players in the case.
Congressional Republicans and the Trump campaign have tried to discredit the dossier as a partisan effort cooked up by the Clinton campaign.More news: British Prime Minister wants to cut all avoidable plastic within 25 years