With many Republicans agreeing with Democrats that raising the debt ceiling is a simple necessity, some think the GOP is left with little negotiating leverage.

But a little is not none: Senate Republicans could threaten to withhold support for raising it and force Democrats to pass it on a straight party line vote. One member of the Senate GOP leadership signaled this is the way the party might go.

"I think we have more leverage on the debt ceiling," Sen. John Cornyn (R-Texas) told HuffPost Friday. "Because unless 51 Democrats want to vote to raise the debt ceiling and all the Republicans vote against it, they’re going to have to give us something in the way of a systemic reform."

The thinking is that in the current political environment, the most vulnerable Senate Democrats up for reelection would run for the hills when confronted with the idea of taking such a politically perilous vote, and that Senate Majority Leader Harry Reid (D-Nev.) will in the end need Republican support to pass it.

Reid has the votes in his caucus to pass the legislation, but only if Republicans do not filibuster it (a filibuster requires 60 votes to be dispensed with). Yet the Senate minority has filibustered almost every major piece of legislation for the last several years, going back to when Democrats were in the minority.

A Cornyn spokesman denied that the senator was talking about filibustering or not filibustering the debt ceiling legislation in order to hang the vote around the necks of Democrats. He pointed out that the senator was referring only to what Republicans are looking for in exchange for their support.

But Senate Minority Leader Mitch McConnell (R-Ky.) has reportedly been talking up the virtues of not filibustering, according to The New York Times. Cornyn’s remark appears to line up with that emerging potential strategy.

Cornyn told HuffPost that Republicans hope to emerge from the debt ceiling showdown having gained three things: a balanced budget amendment to the Constitution, passage of a spending cap bill co-sponsored by Sen. Bob Corker (R-Tenn.) and Sen. Claire McCaskill (D-Mo.), and legislation putting into place some plan to deal with the nation’s debt and deficits.

“Those would be pretty much the universe as far as I understand,” Cornyn said.

The last component could come from a bipartisan group of lawmakers -- the so-called "Gang of Six" -- who are working on a bill that would be based on the recommendations from President Obama’s own fiscal commission last December.

Sen. Rob Portman (R-Ohio) signaled in an interview that there are questions about whether the group –- Democratic Senators Mark Warner (Va.), Dick Durbin (Ill.) and Kent Conrad (N.D.), and Republican Senators Saxby Chambliss (Ga.), Tom Coburn (Okla.) and Mike Crapo (Idaho) –- will be able to hash out an agreement in time to attach it to a debt ceiling bill.

“I think it will be interesting to see whether the gang of six can come up with a legislative proposal, and if they can that’s the likely place where that will be offered,” Portman said.

When Bruce Jostens, the top lobbyist for the U.S. Chamber of Commerce, was asked last week what he saw coming out of the debt ceiling fight, he emphasized budget process reforms and did not mention entitlement reforms as a way of reducing the debt.

“Do I think we are going to raise the debt limit? Yeah, I think we are going to raise the debt limit,” Josten told reporters. “Do I think we are going to have a clean debt limit vote like I think the White House probably wants? No I don’t. I think they are going to have to probably accept some combination of budget process reforms, more spending cuts, other policy riders that restrain spending in certain areas -- some combination of something along those lines.”

All of this ignores what would happen in the House. Even if the debt ceiling increase were to pass through the Senate without any Republican votes, it would fail if the huge GOP majority in the House also voted against it.