SALT LAKE CITY — The rancorous debate in Washington over whether to raise the federal debt ceiling is alarming many of the nation’s governors from both parties, who fear that whatever the outcome, much-needed money will almost certainly be drained from their states.

If the federal debt limit is not raised, several governors said as they gathered here on Friday for the semiannual meeting of the National Governors Association, the ensuing default will harm the economy, make it difficult for states to borrow money and delay some of the vital federal payments that states count on for everything from Medicaid to unemployment benefits.

But even if the debt ceiling is raised, as many governors expect it ultimately will be, states could still pay a high price. Both Democrats and Republicans in Washington want to pair any increase in the debt limit with deep new spending cuts — cuts that many governors fear will hurt their states as they are still recovering slowly from the Great Recession.

“If I can use a whitewater analogy here, the two rocks we need to shoot between is, on the one side, being needlessly driven into default, which will kill the jobs recovery,” said Gov. Martin O’Malley of Maryland, the chairman of the Democratic Governors Commission. “The other rock is massive public sector cuts, by whatever name, that would also kill the jobs recovery.”