House Republicans say Senate Majority Leader Harry Reid is playing head games to undermine their attempt to pass a $659 million border package.

As Speaker John Boehner tries to build support among the Republican caucus for legislation to address the thousands of unaccompanied minors coming to the U.S. from Central America, his Democratic counterpart in the Senate has brought up the toxic issue of immigration reform.

On Tuesday, Reid threatened to attach comprehensive immigration reform legislation to any border bill that makes it through the House — a move that would infuriate immigration hawks and split the GOP caucus.

"If they pass that, maybe it’s an opening for us to have a conference on our comprehensive immigration reform," Reid told reporters. "They are finally sending us something on immigration. Maybe we could do that."

It's unlikely that Reid has the votes to actually do that.

The four Republican members of the “gang of eight” sent a letter on Friday vowing not to support any border bill that includes their comprehensive immigration bill. Senate Republicans are highly unlikely to provide Reid the votes to do this in any event, given their broad opposition to the Democrats' border proposal.

It’s not even clear that Reid would be able to count on all 55 members of his caucus to vote for the “gang of eight” bill, given that a handful of red-state Senate Democrats are vulnerable in the upcoming midterm elections.

That's led many Republican critics to say that Reid is just trying to undermine the House deal before it passes.

“It’s a totally phony threat and I think people need to understand that,” Senate Minority Whip John Cornyn, R-Texas, said. “I think he’s trying to get the House to back down.”

Senate Democrats and House Republicans are at odds over how to address the border crisis.

Senate Democrats want legislation that is more focused on humanitarian relief and closer to the $3.7 billion emergency funding request made by President Obama. House Republicans are more interested in beefing up border security and facilitating the repatriation of the unaccompanied minors sitting in U.S. holding facilities. But even this scaled down package is hardly a sure thing to pass given that most House Democrats are likely to oppose it.

Senate Democratic leaders had been signaling a desire to attach the “gang of eight” bill to the House GOP border bill even before Reid issued his threat. This caused some concern among rank-and-file House Republicans. But sources familiar with the GOP leadership’s whip effort said leaders moved quickly to address concerns, and as of Tuesday afternoon, they felt confident that they had a handle on the issue.

“There will be a blip, I am sure,” said one Republican source. “But the people who would be influenced by such are 'no' votes anyway right now.”

However, cautioned GOP sources, the issue has the ability to cause Rep. Steve Scalise, R-La., the new House majority whip, major problems on his first big bill, and that was apparent at which the speed that Boehner moved to issue a statement responding to Reid and vowing not to consider the “gang of eight” bill — or any other immigration-related policy — if it is attached to the House GOP’s border bill.

The full text of Boehner’s lengthy statement follows:

Senator Reid, embarrassed that he cannot strong-arm the Senate into passing the blank check President Obama demanded, is making a deceitful and cynical attempt to derail the House’s common-sense solution. So let me be as clear as I can be with Senator Reid: the House of Representatives will not take up the Senate immigration reform bill or accept it back from the Senate in any fashion. Nor will we accept any attempt to add any other comprehensive immigration reform bill or anything like it, including the DREAM Act, to the House’s targeted legislation, which is meant to fix the actual problems causing the border crisis. Such measures have no place in the effort to solve this crisis, and any attempt to exploit this crisis by adding such measures will run into a brick wall in the People’s House.

While the White House has abandoned all pretense of governing and the Senate is doing almost nothing to address our struggling economy, Republicans remain committed to addressing the American people’s priorities, and that includes passing a responsible bill this week to help secure our border and return these children safely to their home countries.