House Rules Committee member Rep. Jared Polis (D-Colo.) pressed the latter argument, likening the U.S. budget deficit to a family losing $139,000 per year, then addressing the problem by making sure it only loses $137,000 per year. He noted press reports that said much of the $40 billion in cuts the bill would make come from reclaiming unused funds, and said Republicans rejected Democratic amendments that would have mostly cut military spending, including funds to keep U.S. troops in Europe.

"What is the strategic rationale for an ongoing presence in Germany?" Polis asked. "The Nazis are gone. The Soviets are gone."

Republicans pounced on this and noted that Democrats are objecting to the bill, H.R. 1473, even though it reflects the deal agreed by Congress and the White House last Friday.

"I'd like to remind him … that this happens to be the result of a negotiation that has taken place with three Democrats, the president of the United States, the vice president of the United States, the majority leader of the United States Senate, and one Republican, the Speaker of the House of Representatives," Rules Committee Chairman David Dreier (R-Calif.) said.

"By a three-to-one margin in the negotiating process, Republicans were outnumbered," he added. "So I think that it's a mischaracterization to describe this as somehow a Republican plan that is before us."

House Appropriations Committee Chairman Hal Rogers (R-Ky.) also bluntly replied, saying, "It's being supported by your president. He says pass the bill. It's what we agreed upon."

Rep. James McGovern (D-Mass.), who also sits on the Rules Committee, said explicitly that he would vote against both the rule and the underlying spending bill because it cuts too much from programs related to healthcare, transportation and other areas.

"I'm not pleased … with the so-called compromise," he said. "I urge them to reject this, yet another closed rule, and I urge them to reject the underlying bill."

Democrats also raised a third reason for opposition, which is the requirement under the rule to also consider correcting resolutions that would defund Obamacare and block federal funds to Planned Parenthood.

Rules Committee Ranking Member Louise Slaughter (D-N.Y.) said that structuring the rule in this way raises "serious constitutional questions," as the rule calls on the House not to formally enroll the bill after Senate passage until the Senate has held votes on these two healthcare measures. Earlier in the debate, Rep. Anthony Weiner (D-N.Y.) raised similar questions about whether the House is allowed under the Constitution not to immediately enroll the bill.

Rep. Robert Andrews (D-N.J.) also said it is "troubling" that the rule requires votes on the healthcare measures, but took a position that many Democrats might end up taking: he said he would oppose the rule, but support the underlying bill.

Dreier dismissed all arguments about the link to the two healthcare amendments, noting that the rule does not require the Senate to approve the two healthcare amendments, only to "consider these measures."

"The only thing that we are doing in this rule is ensuring that that agreement is enforced," Dreier added, as a way of noting that both House and Senate votes on the healthcare measures was a condition of last week's agreement on the budget.

The House was expected to vote on the rule to the spending bill later on Wednesday.