The proposal is the GOP counter to a Democratic plan to replace the sequester with more tax revenue collected from millionaires, as well as eliminating agriculture subsidies and reducing defense spending after the end of combat operations in Afghanistan next year.
Under the law that created the forced spending cuts, neither the Pentagon or government agencies can shift money to protect some programs or operations from reduced funding.
While Republicans are divided over how much flexibility Obama should get to avoid the worst impacts of the cuts, they appear unified in opposing any increase in tax revenue to partially offset them.
A January agreement that raised tax rates on top income earners while putting off the forced cuts for two months provided all tax revenue Republicans were willing to consider, party leaders say.
"We can either secure these reductions more intelligently or we can do it the president's way with across the board cuts," Senate GOP leader Mitch McConnell said Tuesday. "But one thing Americans simply will not accept is another tax increase to replace spending reductions we already agreed to."
Obama addressed that viewpoint in his remarks in Virginia, saying there were "too many Republicans in Congress right now who refuse to compromise even an inch when it comes to closing tax loopholes and special interest tax breaks."
"That's what's holding things up right now," the president said. "Keep in mind, nobody's asking them to raise income tax rates. All we're asking is to consider closing tax loopholes and deductions that (Boehner) said he was willing to do just a few months ago."
The full impact of the cuts won't be felt for at least a month, until after a March 27 deadline for Congress to agree on extending funding for the government for the rest of the fiscal year.
An agreement on the overall government funding could soften or eliminate the cuts and much of the hyperbole this week involved posturing for the broader debate of coming weeks.