The Liberal leader is scheduled to present his private member's bill to the House Tuesday night.
It would require board members to meet in public unless they intend to deal with specific issues requiring confidentiality, such as "security, employment, staff relations or tenders," according to the draft text.
Trudeau's bill would also let the board meet in private with the unanimous consent of all members.
Currently, even the dates and times of scheduled meetings are secret and the meeting room is located in an area of Centre Block off-limits to media.
That hasn't stopped Hill reporters from attempting to cover the proceedings when they land in the middle of a high-profile political controversy, however.
Earlier this year, the board came under fire from its own NDP members after rejecting their call to open up its probe of the party's allegedly improper use of House resources to pay for out-of-town satellite offices and mailings.
More recently, the board has found itself under increasing pressure to look into harassment on the Hill.
Liberal whip Judy Sgro has formally asked House Speaker Andrew Scheer, who heads up the board, that it handle the details of any future outside investigation into the allegations against MPs Scott Andrews and Massimo Pacetti. The MPs have been suspended from the Liberal caucus until the matter has been resolved.
Last week, New Democrat Leader Tom Mulcair sent a letter to his Conservative and Liberal counterparts that calls for a more comprehensive approach, including a formal code of conduct, as well as a new officer of Parliament to investigate complaints and a fully confidential process.
Liberal spokeswoman Kate Purchase said there's no conflict between Trudeau's proposal for a more transparent board and the need to keep sensitive matters — such as harassment complaints — private.
"The goal of [Trudeau's bill] is to make the [board] open by default, but of course, in instances where sensitive, personal matters are being discussed, the board would continue to have the ability to go in camera," she told CBC News.
She also pointed out that the Liberals don't want the board itself to investigate the complaints against Andrews and Pacetti, but to "bring in a third-party, neutral investigator."
The board is expected to discuss the matter during its regular bimonthly get-together at 11 a.m. Tuesday.
'A good idea is a good idea': NDP
Meanwhile, despite some lingering anger over how Trudeau dealt with the harassment complaints — specifically, his decision to go public with the suspensions without giving notice to the still unnamed complainants — NDP MPs are expected to support sending his bill to committee when it goes to a vote, which will likely happen early next year.
"We’ll support it," caucus spokeswoman Greta Levy told CBC News.
"Opening up the [board] has been our policy for years now, and I think we appropriately congratulated Mr. Trudeau for adopting our idea a few months ago."
Levy noted that her party had tried to garner the necessary consent for an earlier motion to do the same.
"But a good idea is a good idea," she said.
As for the potentially thorny issue of confidentiality, she pointed out that making meetings "transparent by default does not in any way preclude them being held in camera for good reason."
Still, without at least a dozen or so Conservatives on side, Trudeau's pitch for more open meetings is likely doomed to defeat.
According to Tory whip John Duncan, the government will set out its position during debate on Tuesday.