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The addition of the measure to the House calendar has prompted the White House to get involved. Lobbyists from both sides of the aisle briefed on the issue say the message from President Barack Obama's administration is to sit this fight out and come out in favor of a bipartisan measure later.
"I think what the White House is saying is if it's a Republican-only bill, you shouldn't support it. It's not going to go anywhere," one Democratic K Street source said, speaking on the condition of anonymity. "I don't think the White House is being heavy-handed because nobody thinks this is a real effort."
This lobbyist added that entities that are "trying to do this in a bipartisan way will be looked on more favorably than those who are part of this partisan game Eric Cantor is playing."
The White House press office did not respond to a request for comment.
Some stakeholders who support passing high-skilled worker visa legislation have come out in support of the multiple bills by Members of both parties.
It's a clear example of how advocacy groups and lobbyists are working hard to steer clear of the choppy political waters in the runup to the November elections while still pressing for their legislative priorities.
One organization, the Partnership for a New American Economy, praised all the bills introduced this session: Smith's, Schumer and Lofgren's, plus another by Sen. John Cornyn (R-Texas) that mimics Smith's bill.
"The four sponsors come from both parties and both houses of Congress, showing that there is broad bipartisan support for STEM reform," the group said in a statement issued Tuesday. "The Partnership is optimistic that STEM legislation based on these four bills can be quickly passed by both houses and signed into law by the President."
The Compete America coalition issued a statement Tuesday by Executive Director Scott Corley, hailing the Smith bill but noting, "we look forward to working in a bipartisan fashion to pass critically needed reform legislation that can be signed into law by the president."
Most lobbyists say they expect to see the issue addressed in the lame-duck session, and a Senate Democratic leadership aide said having the bills out now may spur action to resolve negotiators' differences after the elections.
A GOP lobbyist working on the issue said Smith's measure is not likely to pass before the elections despite this week's maneuvering.
"Republicans have offered a really good deal," this lobbyist said. "I think immigration politics is a factor."