There are still procedural and legal hurdles to cross before the files can be released to the public.
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Only after the Senate passes the bill would it head to President Trump's desk to be signed.
There are, however, obstacles to the files being released even if he does so.
The text of the document that will go before the House, for example, notes that the Attorney General could withhold or redact portions of records that contain personal information that "would constitute a clearly unwarranted invasion of personal privacy".
"Simply letting anything out could reveal a lot of private information that's not relevant or appropriate for public consumption," Jonathan Entin, a constitutional law professor at Case Western Reserve University in Ohio, told the BBC.
"There may also be some issues about law enforcement techniques that the justice department might not want to be out there in the public domain," he added.
The proposed House bill also notes that the justice department can hold back any documents that "jeapordise an active federal investigation or ongoing prosecution, provided that such withholding is narrowly tailored and temporary".
That could potentially lead to delays, given that President Trump said in his post that he would be calling for an investigation into Epstein's alleged links with prominent Democrats, such as Bill Clinton and Larry Summers "to determine what was going on with them, and him".
"That's a potential hurdle," Prof Entin said. "If, in fact, this is a serious investigation, presumably the prosecutors will not want everything out there while they're sorting out whether they bring charges."
Doing so, he added, "might create some prejudicial publicity about targets of the investigation" that could ultimately lead to intense litigation if any indictments are brought.
For those who have campaigned for the release of the files, anything short of full disclosure is likely to spark further questions and even outrage.
"I believe the country deserves transparency in these files," Marjorie Taylor Greene, a Georgia Republican now in a public row with Trump over the files, told CNN on Sunday.
"I have no idea what's in the files. I can't even guess," Greene added. "But that is the question everyone is asking is - why fight this so hard?"