Senate Democrats are looking to put President Barack Obama’s $447 billion jobs bill on the floor quickly in an effort to capitalize on divisions among Republicans.

“This is an issue that we’re treating with urgency and this bill will come up in the Senate as quickly as possible,” said one leadership aide. “We think it’s a bold, meaningful proposal that deserves an up-or-down vote.”

Democrats think they can message a likely defeat of the larger plan as GOP obstructionism, and then move on to plucking out smaller items from Obama’s plan, like the payroll tax cut for the middle class and businesses. Because the issue has already split the GOP, it might be harder for Republicans to block those kinds of smaller proposals.

However, Democrats want to give Obama time to sell his plan across the country and build momentum, aides said. Just this week, for example, the president will be in Ohio and North Carolina stumping for the bill.

In the meantime, the Senate will focus on keeping the government operating; Majority Leader Harry Reid (D-Nev.) plans to spend this week trying to avoid shutdowns of the FAA and highway projects and next week trying to pass a continuing resolution to keep the government operating past the Sept. 30 end of the fiscal year.

Republicans, meanwhile, sharpened their rhetoric after Obama went to the Rose Garden on Monday — flanked by veterans, teachers, firefighters and police officers — to introduce his bill.

“I’m sending this bill to Congress today, and they ought to pass it immediately,” Obama said.

The GOP on Monday seized on a briefing by White House budget director Jacob Lew that the administration would pay for the plan with $467 billion in new tax revenue over the next decade, largely by limiting deductions for the wealthy.

“It would be fair to say this tax increase on job creators is the kind of proposal both parties have opposed in the past,” said Michael Steel, spokesman for Speaker John Boehner (R-Ohio). “We remain eager to work together on ways to support job growth, but this proposal doesn’t appear to have been offered in that bipartisan spirit.”

And House Majority Leader Eric Cantor batted away the possibility of the House allowing any tax increases or new stimulus-style spending.

“Anything that’s akin to a stimulus bill, I think, is not going to be acceptable to the American people,” the Virginia Republican told reporters Monday when asked about spending for construction projects outlined in Obama’s plan.

Cantor continued, “Most folks understand that the promises made surrounding the stimulus program were not kept. I don’t believe that our Members are going to be interested in pursuing that. I certainly am not.”

Cantor said the House Republicans’ regulatory reform agenda, which aims to roll back several Obama administration policies, should be included in any jobs package that might be signed into law. He also called for reforming the unemployment benefits program and highway trust fund that Obama is looking to boost with his jobs package.

“I think where people are dissatisfied is the fact that this economy is so bad. And we’re trying to produce results here, and that’s why we’ve taken the position to say let’s set aside our differences and try and transcend them and come together,” Cantor said. “And that’s why I’ve taken issue with this all-or-nothing demand by the White House and the president. It’s just not appropriate and it won’t get us results.”

Cantor wouldn’t say when the proposal might make its way to the floor. Instead, he said, “We will take a look at this; there are provisions which, on the surface, seem to provide an opportunity for us to work together and produce results.”

The senior Democratic aide said they intended to pressure House and Senate Republicans on each of the facets of the jobs agenda instead of letting the GOP cherry-pick ones that they like. “Whose jobs aren’t important? Teachers? … Firefighters? … Veterans?”

The aide said that with the focus on jobs instead of cuts, Republicans are playing on Democrats’ turf.

The GOP risks being painted as “holding up the recovery if they hold up any chunk of it.”

A GOP leadership aide, however, said he doubts Democrats would ever bring the full package to the floor — at least not with proposed tax increases — and they would have a hard time keeping their own troops in line if they did.

“A half-trillion dollar tax hike is an easy way to turn this from a bipartisan bill into a partisan bill,” the GOP leadership aide said.

The aide said Democrats are misjudging the mood of the public — they want Washington out of the way: “I don’t think anybody in America is saying we want another half-trillion dollar plan from Washington.”

Aides in both parties said that while there will be some floor fireworks in the coming months on jobs, the closer the calendar gets to November, the more likely it is that the whole issue ends up getting kicked to the super committee — the powerful joint deficit panel empowered with fast-track rules and which must report out a package before Thanksgiving.

Any short-term jobs items — like an extension or expansion of the payroll tax cut — could conceivably find a home alongside a much larger package of spending cuts, aides said.

But Senate Minority Whip Jon Kyl (R-Ariz.), a member of that committee, ripped Obama’s latest plan and the concept of stimulus in a lengthy floor speech Monday.

Kyl questioned whether government spending could create jobs.

“The government can only give money to a food stamp recipient by taxing that money from someone else or from borrowing that money, and eventually that borrowed money needs to be paid back,” Kyl said. “There’s no free lunch.”