Senate Majority Leader Harry Reid may have announced Tuesday that he won’t block GOP attempts to bring a popular 3 percent withholding bill to a vote, but lawmakers and voters shouldn’t see it as a sign of a thaw in the Capitol’s partisan chill.

The Nevada Democrat said that while he is not inclined to stand in the way, he will try to amend the bill. Such a move could stall the legislation despite its overwhelming passage in the House last week and the backing of the Obama administration.

“The bill that comes from the House, I think we should amend it,” Reid said. “I’ve spoken to the Republican leader; he wants to bring that forward. I’m not going to stand in the way of his bringing it forward.”

Reid’s decision came after days of sniping between Republicans and Democrats. The argument is not over the fundamental policy behind the bill but rather what offset to use to pay for it. According to several GOP and Democratic Senate aides, Reid has insisted on using the offsets in the House bill as a way to fund a transportation authorization bill.

Reid’s resistance to either passing the House bill as is or taking the measure up this week to quickly send it to President Barack Obama has frustrated Republicans.

“The Senate Democratic leadership is paralyzed by devotion to tax hikes and more of the same failed ‘stimulus’ policies,” said Michael Steel, spokesman for Speaker John Boehner (Ohio). “Both the 3 percent provision and the pay-for were specifically supported by the White House. It’s tough to understand why the Senate Democratic leadership doesn’t just pass the House-passed bill. They just can’t take ‘yes’ for an answer.”

Reid countered that the Republicans are protecting “millionaires and billionaires” at the expense of creating jobs for the middle class by opposing the transportation bill.

“I’m terribly disappointed that the Republican caucus has ignored the wishes of the American people,” Reid said.

The bitter back-and-forth is just the latest flare-up resulting from a fundamental breakdown in how the chambers work and how Congress approaches its job, according to aides and lawmakers.

“It’s like walking into a children’s fight,” a Senate Republican aide said Tuesday, explaining that even arcane issues such as pay-fors — which have traditionally been so deep in the weeds that they were smoothed out with little fanfare — have become flashpoints.

“The Senate is not immune to the 24-hour news cycle and bloggers and pressure to stay in the majority or to capture the majority,” Sen. Lindsey Graham (R-S.C.) said. Graham, a former House Member, cast aspersion on both chambers, saying: “It’s a symptom of the Senate becoming more like the House.”

That, Graham and others said, has led to a climate in which neither side is willing to trust the other, which makes it difficult to find common ground — even when it exists.

For months, the House has passed pieces of the GOP’s jobs plan, focusing largely on deregulatory measures that Republicans say will help boost job creation and that are favored by business interests.

But aside from the passage of a package of trade agreements and patent reform, most of the GOP’s agenda has been anathema to Democrats and the White House.

These pieces of the agenda include bills prohibiting new climate change regulations, expanding offshore oil drilling, eliminating federal oversight of some state clean water programs, reducing controls on the disposal of coal wastes and eliminating a series of proposed clean air rules.

None of those bills has a chance of passing the Senate, and the White House has consistently threatened to veto these and other GOP-backed bills.

Dubbed the “forgotten 15” by Republicans, these measures have died in the Senate, where Reid has dismissed them out of hand.

While they have denounced the Senate’s lack of action on the bills, Boehner and House Majority Leader Eric Cantor (Va.) have also acknowledged the need to find common ground with the Senate and White House. The withholding bill is the first in a series of planned mini jobs bills that can draw broad support in both chambers while avoiding veto threats.

That is what makes the back-and-forth over the withholding bill all the more troubling, Republicans and Democrats said.

One senior Democratic Senate aide Tuesday called the atmosphere “just awful.”

A Republican aide agreed, saying, “there’s not a lot of trust” between the two sides at this point, making even relatively benign bills like the withholding measure difficult to move.

Graham warned that lawmakers need a course correction.

“Either the body corrects itself or an outside force corrects the body,” Graham said.

“We owe it to the [country] to not leave behind an institution that in the eyes of the American people is completely destroyed. And we’re coming perilously close to that,” he added.

David M. Drucker contributed to this report.