The push for a balanced budget amendment to the Constitution has become one of the factors in the debate over raising the country’s debt limit, a development that is not exactly unexpected.

Freshman Sen. Mike Lee (R-Utah), one of the founding members of the Senate Tea Party Caucus, said as far back as January that he would seek to filibuster any vote to raise the debt ceiling unless Senate leaders agreed to pass a balanced budget amendment.

Since then, all 47 Senate Republicans have announced their support for such an amendment, proposed by Lee and Republican Sens. Orrin Hatch (Utah), Jon Kyl (Ariz.), John Cornyn (Texas) and Pat Toomey (Pa.). Eleven members of the Senate Democratic caucus and all Senate Republicans in March supported a “sense of the Senate” resolution by Lee on the idea of a balanced budget amendment.

And in recent days, Republicans in both chambers have penned a flurry of op-eds and held a series of events endorsing the idea, culminating in a news conference Thursday afternoon by 10 House Republican leaders, including House Speaker John Boehner (R-Ohio).

But the debate seems unlikely to play any major role in the debt-ceiling talks. (For one thing, the House Republican budget would not be in keeping with the strict limitations of a balanced budget amendment.)

That explains, in part, why leading Republicans such as House Majority Leader Eric Cantor (Va.) and Sen. John McCain (Ariz.) have thus far opposed efforts to tie any debt-limit vote to congressional passage of a balanced budget amendment.

Senate Minority Leader Mitch McConnell (R-Ky.) signed on Thursday to a bill, introduced by Lee, calling for a balanced budget amendment to be included in the debt-ceiling talks. Sen. Lamar Alexander (Tenn.), the chamber’s third-ranking Republican, has signed onto the bill as well.

But McConnell and Alexander, like most other GOP leaders, have declined to sign the “cut, cap and balance” pledge being promoted by more than 100 conservative groups and signed by 12 senators and 36 House members.

Why, then are Republican leaders ramping up their efforts to secure support for a balanced-budget amendment just as the debt negotiations are reaching fever pitch?

From a political perspective, holding amendment votes now – as the House is expected to do next week, with the Senate following soon after – has several benefits for congressional Republicans.

For one, it serves as a “relief valve” for frustrated members, much as a one-week “troop funding bill” did when negotiations on averting a federal shutdown in April entered their final 34 hours.

Such a vote could help members express their desire for a far-reaching plan to stem the country’s rising debt. It could also shield GOP leaders – who would also presumably vote “yes” — from being the target of members’ dissatisfaction at the direction of the ongoing debt-limit talks.

That, in effect, could result in fewer Republicans withholding their support from a final debt-limit deal that lacks a balanced budget amendment.

Holding votes next week also allows GOP leaders to go on the record supporting a balanced budget amendment without going so far as to commit themselves to securing one in the debt-limit talks, as signing the “cut, cap and balance” pledge would do.

House Majority Leader Kevin McCarthy (R-Calif.), who has taken the lead on the balanced budget issue as Boehner and Cantor negotiate with Obama, said the debt limit and the constitutional amendment could collide by early August.

“They are both building for the perfect storm,” he said in an interview this week. “Every time you turn on the news and they’re talking more about the debt of America and they’re talking about the crisis of getting something done, the balanced budget amendment gets stronger.”

The votes may also have the effect of turning up the pressure on Democrats in both chambers to support a balanced budget amendment later down the line.

The amendment requires a two-thirds majority in both chambers, as well as the backing of three-quarters of the states, to take effect. McConnell noted Thursday that Republicans would need the support of 20 Senate Democrats to pass the amendment in that chamber, and House Chief Deputy Whip Peter Roskam (R-Ill.) noted at Thursday’s news conference that dozens of House Democrats would need to sign on as well.

“What we need is President Obama to take a leadership role, to engage members of his party in the House of Representatives to pass this balanced budget amendment language,” Roskam said. “Republicans alone can’t do it. It’s going to take 50-plus Democrats in order to get this done, and we call on President Obama to balance this budget.”

Obama and Democrats adamantly oppose the new version of the constitutional amendment, however, because it contains a provision that requires a super-majority in Congress to raise taxes.

“It totally hamstrings you so that the only two options [to reach a balanced budget] are to end Medicare or Social Security,” Rep. John Larson (D-Conn.) said, suggesting that the version brought forward in the mid-1990s without the tax provision would likely gain the two-thirds majority. “Go back to the 1995 balanced budget and you’d get a lot of Democratic votes.”

The last such vote on a constitutional amendment, in 1997, fell one vote short in the Senate of the 67 required for passage, winning support from 11 Democrats — including then-Sen. Joseph Biden (Del.).

Privately, Republican strategists believe that, once they have the vote on the most conservative constitutional amendment, they could then call the Democrats’ bluff and vote on a version of the constitutional amendment without the tax provision.

Republicans could also use the votes to build momentum for their case to supporters that in 2012, retaining control of the House — and retaking the Senate — will be key to achieving their policy goals. (Bolstering that argument is the fact that a balanced budget amendment would not require the president’s signature in order to take effect.)

All of this means that next week’s votes could, in fact, have a host of political benefits for congressional Republicans. Holding much sway over the direction of the White House-led debt negotiations, however, isn’t likely to be one of them.