"I certainly join the chorus of voices that are calling for that administration and members of Congress to come together and find more responsible cuts," said Indiana Governor Mike Pence, a Republican who served in the House of Representatives during the 2011 negotiations that led to the cuts.
Congress is expected to return to work on Monday to try to forge an agreement on avoiding the cuts, which are known as sequester or sequestration and are intended to save $1.2 trillion over 10 years. At the end of 2012, lawmakers decided to push the sequester's start date back to March 1 from Jan. 1.
Hawaii Governor Neil Abercrombie, a Democrat who served in the House of Representatives for nearly 20 years, anticipates they will decide again to delay the start of the process.
"I think there will be a push-off," he said on Saturday. "But what will happen is some deep breaths will be taken, some real soul-searching will take place about what the political implications are going to be if this happens again and again. And I think a more long-term resolution will take place."
Republican and Democratic governors agree the federal government must shrink the deficit. They have already met with the White House and lawmakers in the first two months of 2013 to discuss sequestration alternatives.
In those meetings "We said, look we all know we've got to suffer a little bit together to get this ship righted," said Arkansas Governor Mike Beebe, a Democrat. "We're willing to take our share, but we don't want to take more than our share."
They are asking that any plan addressing the deficit should also help states save money, give them flexibility in deciding which areas to cut in their states, and not shift costs onto their budgets, said Governor Mary Fallin, a Republican from Oklahoma.
Sequestration was designed as a threat by Congress, which set itself a deadline in the fall of 2011 to agree on spending reductions. Because it missed the deadline, a scheme of automatic cuts decided by formula kicked in.
"We should all remember that sequestration was originally designed by Congress as something so odious, so repellant that it would force both sides into a compromise," said Governor John Hickenlooper of Colorado, a Democrat. "There can be no question that this is something nobody wants."
During and after the 2007-09 recession, states' revenues plummeted to lows not seen in decades, while newly jobless and homeless residents turned to the their governments for help. Only in the last year have their revenues returned to pre-recession levels and spending demands eased. Many governors see sequestration as a threat to their slowly improving economies.
The two main sources of federal money for states - Medicaid, the healthcare program states run with federal reimbursements, and highway funding - are exempt from sequestration.
The remaining programs that are subject to sequestration directly provide on average 6.6 percent of states' revenues, according to Pew Center on the States. In South Dakota, they provide 10.3 percent.
The federal government, by employing people and buying goods and services, also plays an indirect role in states' economic fortunes.
On average, federal spending on procurement, salaries and wages within a state represents 5.3 percent of its gross domestic product, according to Pew. For Virginia, Washington, D.C. and Maryland, federal spending has the most economic impact, making up 19.7 percent of their GDP.
For federal defense spending, the average is 3.5 percent of state GDP, with Hawaii having the highest at 14.6 percent.
An analysis by Wells Fargo Securities Economics Group last week found that under sequestration states close to the nation's capital and in the South "will be the hardest hit, while states in the Midwest and the West Coast will likely be impacted to a lesser extent."
Governor Peter Shumlin, of Vermont, who also heads the Democratic Governors Association, called the reduced direct funds and potentially harmful economic effects a "double whammy."
"I have no idea what they're going to do down here," he said about Congress reaching an agreement. "All I can tell you is that the President ... seems quite convinced they're actually willing to push the button and undermine the economic recovery and prosperity that we're seeing in our states."
Democratic and Republican governors alike say statehouses and businesses have been left with too much uncertainty as the country approaches sequestration. Federal agencies are only now disclosing where they will cut spending and reduce work forces.
Fallin has pushed since the summer for an alternative and also does not want sequestration repeatedly postponed.
"Do the least harm and give us the flexibility," she said. "It is not good to have the sequester talk every couple of months."