White House Chief of Staff Denis McDonough said Sunday that Presiden Obama has not given up on the prospect of averting the looming across-the-board spending cuts known as the sequester.

"Well we’ve not given up on this, Bob, and the reason we haven’t given up on it is because it’s going to have a real impact on middle class families," McDonough told Bob Schieffer on CBS News' "Face The Nation."

Senate Democrats proposed a plan last week to delay the threat of the deep cuts -- which are set to kick in March 1 if lawmakers do not act -- in part by raising taxes on millionaires. The White House issued a statement in support of the proposal, but top Democrats acknowledged that it faces tough odds.

Republicans have mainly opposed discussing new tax revenue, saying the matter was addressed in the deal to avert the "fiscal cliff," which included tax rate hikes on the wealthiest Americans.

"When you look what at Senate Democratic plan and the president’s plan – both very balanced plans that get some savings in this deficit fight from spending cuts, and some savings from increased revenues," McDonough said on "Face The Nation." "What our friends in the House have told us is that they will not even consider anything that includes increased revenues. Not even closing loopholes for corporate jets; closing loopholes for oil and gas companies."

Sen. John McCain (R-Ariz.), who last week expressed a possible willingness to consider supporting new tax revenue to avoid sequestration, on Sunday pressed Obama to negotiate with Republicans, and reiterated his openness to consider new tax revenues.

"I said that I would be glad to close some loopholes about these kinds of subsidies that are outrageous and disgraceful," McCain said, as he called on Obama to step up his leadership on the matter of sequestration.

On CBS, McDonough also underscored Obama's deficit reduction plan, saying: "Over the course of the last several months we’ve got agreement on $2.5 trillion in deficit reduction over the next 10 years. We’re ready to do another trillion-and-a-half to get to the $4 trillion mark that every economist in the country says we need to do to stabilize the debt problem."

House Budget Committee Chairman Paul Ryan (R-Wis.), who has said he thinks the sequester will happen, explained why on ABC's "This Week With George Stephanopoulos."

"The Senate hasn't passed a bill to replace the sequester. The president gave a speech showing that he'd like to replace it, but he hasn't put any details out there.  So that is why I conclude I believe it's going to take place," Ryan said.