"It’s already transcribed, it’s ready, it’s just a matter of printing it out," Conaway said, as quoted by the Washington Examiner newspaper.
The decision comes days after the US Senate Intelligence Committee released its transcript from an interview with Simpson.
Many Republican lawmakers believe that the dossier, prepared by former British spy Christopher Steele in a project reportedly funded by Democrats during the 2016 presidential campaign, formed the basis of an FBI decision to investigate allegations of Trump campaign collusion with Russia.
Russia has repeatedly denied any collusion, while the dossier has been repeatedly criticized as being based entirely on unsubstantiated rumors and therefore inappropriate as the basis of an FBI investigation.
The FBI probe was taken over by Special Counsel Robert Mueller after Trump fired the agency's director, James Comey. Multiple congressional committees are also investigating the collusion charges.
In the course of the investigation, Clinton and Trump have been exchanging mutual accusations in short-circuiting the democratic process and the alleged Russian collusion.
Most lately, the US President has declined to commit in an interview with special counsel Robert Mueller, who is leading an investigation into the alleged collusion between Trump and Russia, a probe the president has repeatedly called a "witch hunt" and "a stupidity that Russia and the whole world is laughing at."