The transcript was expected to be released later Thursday.
Fusion GPS, based in Washington, was hired by a law firm that represented Hillary Clinton's campaign and the Democratic National Committee. Fusion GPS in turn hired former British spy Christopher Steele to investigate Donald Trump's business dealings with Russia.
The move comes on the heels of the release by Democratic Senator Dianne Feinstein of Simpson's testimony before the Senate Judiciary Committee. The panel's Republican chairman, Chuck Grassley, had not agreed to the release.
Trump slammed Feinstein for the release, calling her "sneaky Dianne," and calling the release "underhanded" and "possibly illegal," a claim that legal experts and lawyers dismissed as untrue.
In the Senate transcript, Simpson said Steele uncovered "alarming" evidence of collusion between the Kremlin and Trump's team and that he gave the dossier to the FBI because he was "very concerned" about a potential national security matter.
Trump has repeatedly criticized the dossier, which was based on Steele's investigation, calling it "bogus" and "discredited and phony." The president also called the Russia probe the "greatest single witch hunt in American history" and urged congressional Republicans to "finally take control" of the investigation.
Feinstein said Simpson requested the transcript of his testimony be released to the public and that the American people deserved the chance to see his words and judge for themselves.
While special counsel Robert Mueller is investigating ties between Russia and Trump's inner circle on behalf of the Justice Department, House and Senate investigations were also launched.
Earlier this year, the U.S. intelligence community released a report that stated Russia had meddled in the 2016 election, showing a preference for Trump over Clinton, his opponent. Russia denies meddling in the election, and Trump has denied any collusion.