The council commissioned high court judge Jeff Blackett, the RFU's disciplinary chief, to undertake a review of the process which led to Steele's departure after just nine months.
One of the recommendations contained in the 52-page report is believed to have said that then-chairman Martyn Thomas and the majority of the board should resign.
The report was considered at the last board meeting on July 10, at which Thomas did stand down - although he insisted that was only to concentrate on his role as acting chief executive.
The council voted against publishing the report after Thomas' lawyers threatened to sue Blackett for defamation during the meeting.
A vote of no confidence in the board was also defeated at the meeting by a majority verdict.
As a result, the report was kept confidential - but since then at least 24 council members are understood to have signed a motion, which has been submitted to the RFU, demanding publication.
The Daily Telegraph has reported that all council member received a letter outlining the possible ramifications of publishing Blackett's findings.
The newspaper claims that legal opinion was given by Deputy High Court judge Robert Englehart, QC, who concluded that publication would be "inadvisable".
The RFU has been engulfed in turmoil since Steele was ousted from his post on June 9 after losing the confidence of Thomas and the board.
Central to Steele's downfall was the RFU's botched attempts at recruiting a performance director.
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