BOSTON (AP) — Governors hamstrung by the sluggish economic rebound in their states and bound to balance their own budgets are pressing anew for Washington to step up with more help, some say even if it means adding to the nation’s red ink.

Republicans and Democrats alike wrestled with how to capitalize on a fledgling rebound as they talked dollars and sense at their summer meeting just days into a new budget year and as the economy shapes dozens of gubernatorial races across the country.

‘‘All states still are facing tough fiscal situations even though I do believe we’re in recovery,’’ said West Virginia Gov. Joe Manchin, a Democrat who’s taking over as chairman of the National Governors Association.

Added Gov. Jim Douglas, R-Vt., the outgoing chairman: ‘‘Governors have done what is necessary to get through this’’ — repeatedly cutting budgets, restructuring government, laying off workers and draining rainy day funds.

But both men said states can’t continue to climb out of the recession alone, and the NGA renewed its bipartisan appeal for Congress to pass stalled jobs legislation that includes billions of dollars in aid to states.

Just days before the new budget year began July 1, the House and Senate failed to complete legislation that would have extended, through June 2011, important parts of the federal stimulus program enacted last year to provide unemployment insurance and help offset recession-driven cuts to education, health care and public safety.

The measure offered $35.5 billion for unemployment benefits for the long-term jobless and $16 billion for Medicaid, the public health care program for the poor. It also would have added an estimated $33 billion to the deficit.

Even so, several Democratic governors suggested in interviews and during panel discussions that the short-term gain was worth the long-term pain. But some Republican governors were more tentative in their support, suggesting any aid should be offset by paying for it from other areas of the budget.

In February, 47 governors sent a letter in February to Congress requesting lawmakers give states more money for Medicaid, and NGA leadership renewed that call as the three-day gathering opened. Said Douglas: ‘‘We’ve taken a pretty clear position on it. We’re for it.’’ Still, he acknowledged disagreements over how to pay for it and added: ‘‘We need to find that common ground.’’

Republicans privately said NGA didn’t send a new letter during this meeting because support in the GOP ranks had thinned over the last several months. The party is facing angry tea party advocates demanding less spending.

Gov. Bob McDonnell, R-Va., said he worries about deficits. ‘‘There’s a growing sentiment among the citizens of this country that the rate of spending and the increase of spending at every level of government is unsustainable.’’

And Wyoming Gov. Dave Freudenthal, a Democrat, said money from Washington wasn’t the answer. ‘‘Each of us has to figure out a way to pare back the public’s expectation of what government’s going to pay for.’’

But Gov. Pat Quinn, D-Ill., said: ‘‘We need more help from Washington to protect against job cuts and health care cuts. If we don’t do that, we’re following Herbert Hoover economics.’’

Maryland Gov. Martin O’Malley, a Democrat, said that without Washington’s assistance, ‘‘the danger of a double-dip recession is greater.’’ Democratic Gov. Brian Schweitzer of Montana, called for ‘‘a little less talk and a lot more action’’ from Washington.

Governors headed to Boston for the weekend ready to share ideas on a range of issues, from health care to alternative energy to education. But the economy dominated the discussions.

States have suffered for two straight years as collections from sales, income and property taxes came in lower than expected when people stopped spending money, lost their jobs and the housing market tanked. Governors also saw demand for state services soar.

While the federal government reports signs of economic growth, a report last month by the NGA and the National Association of State Budget Officers found that state fiscal conditions continue to deteriorate.

States are expected to have faced $296.6 billion in shortfalls between the 2009 and 2012 budget years, and the report found that the new budget year will be just as challenging, despite the expectation of modest increases in tax collections.

There are political realities to the states’ economic challenges.

This fall, 37 states will elect governors; a majority of those races already are competitive. The economy is dominating contests across the country, and Democrats are on defense in a slew of economically struggling states.

Among the most vulnerable Democrats is Iowa Gov. Chet Culver, who plans to spend the next four months telling voters that ‘‘there are some bright spots, and beyond the horizon it’s really bright because we’re weathering this economic storm.’’

Still, he said he’d welcome money from Washington to ensure the recovery continues.