By Lauren French and David Alexander

WASHINGTON (Reuters) - Former Vice President Dick Cheney told Republican lawmakers on Tuesday that looming defense cuts could have a serious impact on the U.S. military, even as a new analysis predicted the budget reductions that begin in January could cost 2 million jobs.

Lawmakers on Capitol Hill fretted over how to resolve the issue but remained dug into their longstanding positions. Democrats are calling for increased revenues as well as further budget cuts and Republicans are insisting on spending reductions alone and no new taxes.

Cheney's visit to the Capitol came as mayors of two major U.S. cities warned that the looming $1.2 trillion federal budget cut is the biggest threat to their local economic recoveries. The 10-year, across-the-board cut is due to go into force on January 2 under a process known as sequestration.

Cheney, who was defense secretary during the first Gulf war and presided over the post-Cold War military drawdown, told Republican lawmakers in a closed-door session the problem should be fixed but made no specific recommendations, said Senator Lindsey Graham.

"I thought he did a very good job (explaining) about how he reformed the Pentagon, eliminated some weapons systems that were wasteful ... (and) basically impressed upon us that the sequestration that we are about to enter is just devastating to long-term planning and readiness," Graham said.

Phoenix Mayor Greg Stanton, speaking at a defense industry event, said Arizona stood to lose 50,000 high paying aerospace and defense jobs as a result of "Congress's failure to deal with looming indiscriminate cuts" to the defense budget.

"That is the Number One threat to our local economy," Stanton said. "And it would put our city, and the state I represent, back to recession. It is exactly the wrong direction for us to go in."

Mayor Jerry Sanders of San Diego, where defense is one of the major industries, said the cuts would lead to a loss of confidence, and once that happens "our economy is going to go back to right where we were and cities have nowhere to go but have to cut vital services."

Congress passed the Budget Control Act last year to try to impose fiscal discipline on a federal government that has been running trillion-dollar deficits and has amassed a $15.8 trillion debt.

The measure called for the Pentagon to reduce its projected spending by $487 billion over the next decade and trimmed non-defense spending by a similar amount.

The law also established a congressional panel to find another $1.2 trillion in budget cuts over a decade, and ordered automatic across-the-board spending reductions if the group failed to reach a compromise by the end of 2011. Lawmakers were unable to agree on a deal and now face the automatic cuts.

Defense Secretary Leon Panetta and others have warned the cuts would be devastating and would undermine the Pentagon's new defense strategy.

But some budget analysts note that the pending cuts follow a decade of increases in defense spending and are far smaller than during previous drawdowns in military spending. Even if the January 2 cuts do take place, the Pentagon budget will still be comparable in size to its 2006 spending plan.

ECONOMIC IMPACT

Stephen Fuller, an economist at George Mason University, released an analysis prepared for the Aerospace Industries Association saying the budget cuts due to go into force on Jan 2 will cost 2.14 million jobs and reduce gross domestic product by $215 billion.

The cuts would reduce spending by about $1.2 trillion over a decade. Fuller said the first year's cut would slash Pentagon spending by $56.7 billion, resulting in 325,693 direct job losses as well as another 764,666 indirect job losses.

The government's cuts in non-defense spending would have a similar impact. The $59 billion cut to non-defense spending would directly cost 420,529 jobs, which in turn would trigger another 626,820 indirect job losses, Fuller said.

"The results are bleak but clear-cut," he said. "The unemployment rate will climb above 9 percent, pushing the economy toward recession and reducing projected growth in 2013 by two-thirds."

The new analysis comes a day before the House of Representatives is due to vote on a bill requiring the Obama administration to provide details of what programs would be cut under the January 2 spending reductions. Representative Steny Hoyer, the No. 2 Democrat in the House, said the measure was expected to pass.

A similar measure has passed the Senate, but it is unclear if the two chambers will ultimately agree upon a final version that can be sent to President Barack Obama to be signed into law.

Senator John McCain, the ranking Republican on the Senate Armed Services Committee, blamed Obama for the situation, saying he had failed to work with Republicans to find a way to resolve the problem.

"The facts are devastating, the American people should understand that. Then maybe they'd put ... pressure on the president to sit down and talk with us and help us work this out and avoid it," McCain said.

(Additional reporting by Thomas Ferraro and Susan Cornwell.; Writing by David Alexander; editing by Christopher Wilson)