Sen. Rob Portman, a member of the Joint Committee on Deficit Reduction, says the panel should balance public meetings and private conversations between Members and with staff in order to make progress.
Sens. David Vitter (R-La.) and Dean Heller (R-Nev.) separately wrote to the committee urging it to adopt a sunshine rule Thursday that would require any meetings of the committee with a quorum of members to be public and televised. Another rule would require prompt disclosure of campaign contributions to committee members.
"It's just plain good government for the public to know what special interests are trying to influence the committee," Vitter said in a statement.
The committee also faces pressures both on what to do and what not to do as members must quickly try to agree on an outline for their deliberations. Democrats on the panel have already been pushing for a jobs package in the short run with Van Hollen tweeting about the need for infrastructure spending to put people back to work alongside longer-term deficit reduction. Democrats argue that the deficit problems cannot be solved unless unemployment drops and that cutting too quickly would make the economy worse.
GOP leaders, meanwhile, huddled with their picks Wednesday and expressed optimism that the panel would at least meet the requirement for a $1.2 trillion deficit reduction package and perhaps more.
"We're committed that this committee get an outcome," Speaker John Boehner (Ohio) said after the meeting. "We believe that by reducing our debt we create a better environment to create jobs in America. ... And we're encouraging our Members to work hard to get an outcome."
"Failure is not an option," Senate Minority Leader Mitch McConnell (R-Ky.) added. "This committee is going to succeed in meeting at least the floor that's established in the legislation."
Senate Majority Leader Harry Reid (D-Nev.) said he's not going to try to "micromanage" the committee and is encouraged by leaders talking about the importance of avoiding the $1.2 trillion spending-cut trigger that would take effect starting in 2013 if the committee fails to reach agreement.
The committee also faces an early test on whether it will tackle the complicated issue of tax reform, which could be the linchpin of whether the panel can reach a grand bargain that truly deals with the long-term fiscal crisis for the coming decades.
Several Senators not on the committee said Wednesday that there simply isn't time to do a full tax reform package by Thanksgiving. Sen. Orrin Hatch (R-Utah), the ranking member of the Finance Committee, said he wouldn't want the committee members to do tax reform by themselves even if they could.
The complexity of tax reform had some Senators, including Democrat Ron Wyden (Ore.), speculating that the super committee could devise a future fast-track process, setting the stage for tax reform at a later date.
Humberto Sanchez, Jonathan Strong and David M. Drucker contributed to this report.