Glenn Simpson, a founder of the research firm Fusion GPS, spoke to investigators with the Senate Judiciary Committee for 10 hours in August.
"The innuendo and misinformation circulating about the transcript are part of a deeply troubling effort to undermine the investigation into potential collusion and obstruction of justice", said Dianne Feinstein, who is the senior Democrat on the Senate Judiciary Committee.
"The only way to set the record straight is to make the transcript public", she said.
In recent weeks, as the political fights about the Russian Federation investigation and the dossier have intensified, Simpson has urged the committee to release the full transcript of his interview, arguing that Republicans are trying to obscure, rather than reveal, what happened in 2016.
Mr Trump has repeatedly denied colluding with Russian Federation during the campaign, and called the entire investigation a "witch hunt".
Mr Simpson also said Mr Steele had cut off contact with the FBI after former agency head James Comey announced he was reopening an investigation into Hillary Clinton's private email server. Steele's reports were eventually compiled into a dossier alleging the Trump campaign coordinated with the Kremlin - a claim the president has repeatedly denied.
"Essentially what he told me was they had other intelligence about this matter from an internal Trump campaign source", Simpson said.
Grassley's office slammed Feinstein for releasing the testimony, saying it came as the panel was still trying to secure testimony from other witnesses, including Trump's son-in-law, Jared Kushner.
Simpson said Steele's concern, "which is something that counterintelligence people deal with a lot, is whether or not there was blackmail going on, whether a political candidate was being blackmailed". In a statement, Fusion GPS commended Feinstein "for her courage" in releasing the document.
The disagreement between Grassley and Feinstein is further evidence of a breakdown on the panel after an initially bipartisan investigation.
Simpson and Fusion co-founder Peter Fritsch had urged the committee's Republican leaders to release transcripts of their testimony in a New York Times opinion piece last Tuesday entitled "The Republicans' Fake Investigations". Republican Sen. John Cornyn of Texas told reporters that the release was a "good idea" and that transparency is important as they work to understand the impact of the dossier.
A spokesman for Grassley, Taylor Foy, called it "confounding" that Feinstein released the transcript "unilaterally", over the objections of the Republican majority.
Grassley and Republican Sen.
"This was like, you know, you're driving to work and you see something happen and you call 911", Simpson told investigators.
Democrats on the committee said they were not consulted ahead of the Republicans' referral to the Department of Justice.