Senator Malcolm Byrne, a member of the Oireachtas Media Committee, has said Ryan Tubridy and his agent Noel Kelly must "provide pieces of the jigsaw" and explain why they did not correct undeclared payments in RTÉ's figures.
Mr Tubridy and Mr Kelly will appear at two back-to-back Oireachtas committee hearings on Tuesday morning.
They will appear before the Public Accounts Committee (PAC) at 11.30am before a scheduled 3pm appearance before the media committee.
Mr Byrne told BreakingNews.ie: "Everybody talks about the figures of the top presenters, and clearly some people knew the figures published were incorrect, and that Ryan Tubridy had a side deal, and the question is why did nobody put their hand up to correct that?"
The Fianna Fáil Senator added: "We have a role to try in as far as we can to complete a full picture as to what happened. We have certain pieces of the jigsaw, we believe that Ryan Tubridy and Noel Kelly can provide other pieces of the jigsaw.
"It will be interesting to hear their opening statements before both the Media Committee and the Public Accounts Committee to the extent that they will answer a lot of the questions around what has happened.
"The committees will be time-limited to three hours in each case. It is very much focusing on the issue of the undeclared payments to Ryan Tubridy and the deals entered into between Noel Kelly and RTÉ management."
Mr Byrne said Mr Tubridy and Mr Kelly will have questions to answer over their failure to correct the record over undeclared payments to the presenter, which sparked the pay scandal at the national broadcaster.
"I certainly think the details of that arrangement from Noel Kelly and Ryan Tubridy's perspective will need to be outlined but also why neither Ryan Tubridy nor Noel Kelly sought to correct the record when the salaries were being published. They knew at the time the Oireachtas and the public were being misled, but they did not look to correct the record."
Mr Byrne praised the initial work of new director general Kevin Bakhurst, who has announced changes with a temporary leadership team.
He also expressed hope that former director general Dee Forbes and head of content Jim Jennings will take up invitations to appear before the Oireachtas committees when they recover from illness.
"On their recovery hopefully they might consider doing that, or the committee is happy to take a written submission from either or both of them," he added.
The previous appearances from RTÉ executives and board members ran way over time, with some TDs and senators interrupting and asking long and drawn out questions.
Mr Byrne feels the vast majority of his colleagues have acted professionally, however he said there were some members "grandstanding".
"I think most members, TD and senators, do not grandstand. There are a few who tend to grandstand and who were more interested in hearing the sound of their own voice, not the responses. You can be tough in your questioning, but you should still be respectful of the witnesses and I think the majority of members were.
"The sessions are being limited and priority is being given to committee members, I fully anticipate my colleagues will stick to the issues. The key around all of this is trying to elicit as much information as possible, I think the committees have done a good job of that to date.
"The big challenge will be piecing all the jigsaw together, getting the transparency and there is accountability and action taken for those responsible. But I think once this week is over, then it's important to let the teams the minister has sent into RTÉ looking at culture and governance to do their jobs, as well as forensic accountants to examine the books.
"We do view public sector broadcasting as important. It's central to a democracy and important to hold those in positions of power to account but RTÉ need to practice what it preaches and ensure it is also transparent and accountable... particularly with the expenditure of public monies."
Mr Byrne said the "drip effect" has been damaging for RTÉ.
He added: "I've called for a register of interests in RTÉ for some time, I sought that when executives were before our committee, and I'm glad the new DG has acted quickly.
"For instance, if someone is doing a segment on the motor industry, and they have a side deal with a car company we are entitled to know about it. No different to how we as politicians have to declare and register our interests, I think presenters, producers and senior RTÉ personnel should do the same."
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