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Washington (CNN) -- House Speaker John Boehner is again facing strong opposition from a group of conservatives who are trying to derail his strategy as time runs out to avert a shutdown.

They are ripping his plan to kick the fight about the President's executive order on immigration until next year, arguing that voters who supported Republicans in last month's midterm elections expect Congress to wage the battle now.

The resistance was expected, but it's not expected to be fatal to the bill. Boehner and top House GOP leaders knew that any plan short of barring all funding for federal agencies that issue new visas would be rejected by hardliners. While the group is loud, and is demanding multiple changes in return for their support, multiple House Republican aides and members are skeptical it will cause the bill to fail.

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But with the House vote likely coming next Wednesday - just a day before the government is slated to run out of money - the continued push and pull between conservatives on both sides of the Capitol and their leaders will keep the drama high right up until the last minute.

On Tuesday, House Republican leaders outlined a bill to fund most federal agencies for a full year. But the bill would only approve money for the Department of Homeland Security - which issues new visas - through March.

In addition, House Republicans scheduled scheduled a vote for Thursday on a measure authored by Florida GOP Rep. Ted Yoho, a conservative who often breaks with leaders, that explicitly calls the President's immigration actions unconstitutional.

Texas Republican Sen. Ted Cruz -- who helped coordinate conservatives ahead of last year's government shutdown -- made the unusual move Wednesday of crossing over to the House side of the Capitol to join a press conference with a handful of House conservatives.

Cruz didn't mention Boehner's name, but his presence with about a half dozen House members was essentially a stick in the eye to the Speaker. The Texas Senator made his opinion clear on the latest House GOP plan telling the crowd he opposed "having a meaningless show vote."

The move clearly frustrated allies of GOP leadership in the House. When asked about the rally, North Carolina GOP Rep. Rene Ellmers told CNN flatly that "Sen. Cruz needs to stay in the Senate."

Referring to last year's shutdown that many credit Cruz with instigating, Ellmers added: "I think Sen. Cruz wants to fan the flames here but I think everyone here has become more savvy to his ways."

READ: Ted Cruz causes headaches. Again.

Senate Republican leaders and their aides expressed confidence the House bill would get enough GOP support to pass the bill and send it to the Senate. They adamantly oppose a government shutdown and the political damage they fear it would do just as Republicans are taking control of the chamber.

"There's not going to be a government shutdown," incoming Senate Majority Leader Mitch McConnell told XM Sirius radio on Wednesday. "Obviously, we think the President has been way out of line with his executive amnesty and we certainly don't approve of that. But, you know, shutting down agencies of the government that have nothing to do with immigration serves no purpose other than to rattle the economy."

Those House Republicans opposed to the bill say along with Cruz, other Senate conservatives like Jeff Sessions of Alabama, Mike Lee of Utah and David Vitter of Louisiana are pressing for the House to take a bolder stance.

Some of these conservatives are complaining that this latest plan from Republican leaders doesn't match the rhetoric they used after the President laid out the details of his plans on immigration last month.

"It's hard for me with every major figure in our party -- starting with Reince Priebus coming down to the lowliest of members like myself -- saying we are going to do everything we can to stop this and to do nothing to stop it really hurts," Republican Rep. Mick Mulvaney of South Carolina told reporters Wednesday.

But House Republican aides say that discussions among members aren't turning up opposition from large numbers of members outside the group that typically splits with leaders on annual spending bill votes.

Texas Republican Rep. Bill Flores, who was recently elected to head up an influential group of House conservatives, told CNN late Tuesday he thought what Boehner outlined was "a good strategy."

"We are where we are, and it's still a decent way to handle it and still provide some ability to check the President later on," Flores said.

READ: 'Cromnibus' looms over Obama-McConnell meeting

Georgia Republican Rep. Doug Collins was among a group of conservative members who met with House Majority Leader Kevin McCarthy on Wednesday morning. He told CNN he is still reviewing the details, but is open to supporting the leadership's plan.

"You fight strategically," Collins stressed, noting he still serves in the Air Force. He said despite GOP Senators like Cruz and other outside groups urging a major effort to strip out money for the executive action, the reality is that until next year, Democrats still control the Senate so it's important to look "at the long term battle."

It's possible some tweaks are made to the plan. There is some discussion about moving the date on the provision in the spending bill that funds the department of Homeland Security to late January, or sometime in February so that the newly sworn in GOP Senate and House can try to work on a strategy earlier to challenge the immigration plans.

Ellmers predicted that Republicans will still need some Democrats will back the measure. That's been the case with every annual and short term spending bill that has come to the floor since Boehner became Speaker.

House Democrats are holding back on how they will approach the vote - saying they are still waiting on final details. On Tuesday, House Democratic Whip Steny Hoyer called the GOP plan "a game" and "bad policy," but declined to say he would urge his colleagues to oppose it. Senate Democratic Leader Harry Reid suggested he could support the plan.

For now, Democrats are once again enjoying sitting back and watching the public feud among Republicans.