By Lesley Wroughton and Jonathan Landay

WASHINGTON (Reuters) - As U.S. Secretary of State John Kerry met with Russian President Vladimir Putin in Moscow on Thursday to propose close military coordination in Syria, the White House and Pentagon offered lukewarm support for the plan and demanded that Moscow show it was serious about fighting Islamic State, not just propping up President Bashar al-Assad.Kerry himself took a tough line after his meeting on Thursday with Putin, State Department spokesman John Kirby told reporters in Moscow.

"Secretary Kerry emphasized that absent concrete, near-term steps, diplomatic efforts could not continue indefinitely," Kirby said.

Kerry's proposal would create a new center where U.S. and Russian militaries would share intelligence and coordinate air strikes against Islamic State and the Nusra Front, an al Qaeda affiliate. The move has angered U.S. military and intelligence officers and diplomats who argue that Russia has proved repeatedly that it cannot be trusted.

Pentagon spokesman Peter Cook said Defense Secretary Ash Carter supported Kerry's efforts to encourage Moscow to "do the right thing" in Syria. But he added that Carter has experience dealing with Moscow and maintains a "healthy dose of skepticism" about Russia's aims in the conflict.

    "The secretary supports Secretary Kerry's effort ... but he has also said that he's had questions about the Russian activities up to this point," Cook told reporters.

    "If the Russians are prepared to do the right thing in Syria, then the secretary of defense would be open to that conversation," he said. "But we're waiting to see what's going on. ... It's not clear that we'll ever reach an agreement." WHITE HOUSE SAYS IT'S UP TO RUSSIA

At the White House, spokesman Josh Earnest's response was similarly cautious.

"It's time for Russia to make serious decisions about how they want to use their influence inside of Syria both when it comes to their diplomatic influence, but also when it comes to their military capabilities," Earnest told a daily briefing on Thursday at the White House,

"They have to decide if they're going to use the military to prop up the Assad regime or if they're going to use their military to go after extremists," Earnest said, "Our case is you can't do both."

Kerry has expressed growing frustration with the lack of progress in ending Syria's five-year civil war despite agreements with Russia to secure a lasting nationwide cessation of hostilities and ensure that humanitarian aid reaches besieged communities. U.N.-led efforts to bring the warring sides together to negotiate a political transition also have failed.

State Department spokeswoman Elizabeth Trudeau acknowledged a "very robust and very vigorous conversation" among government agencies over Kerry's plan.

Asked whether Kerry's visit had the full backing of the White House, she added: "I'm not going to characterize that for the White House, but I would say that the secretary is going to ... have discussions in Moscow that present the administration's views."

Anger Within the Administration    But U.S. military and intelligence officers and diplomats involved in Syria policy expressed anger about Kerry's proposal, which several called naive.

    The Russians, said one official, had known for weeks about Kerry's upcoming visit to Moscow, yet two days before he was due to arrive they bombed a remote camp near Syria's border with Jordan used by U.S.-backed rebels.

"That is consistent with what the Russians have been doing all along," said a U.S. military official, who like others spoke anonymously to criticize administration policy. "More of it has been to us, not with us, and the message has been clear all along: 'When our interests differ from yours, ours win."

"They've violated every cessation of hostilities agreement they’ve ever agreed to, sometimes within hours," the official said.

    Another American official, who has helped train members of the Free Syrian Army rebel alliance, said the U.S.-backed opposition forces have been asking why the administration keeps seeking Russian help and are growing increasingly bitter, their anger compounded by what they complain is a lack of American military support.

Former U.S. Ambassador to Syria Robert Ford expressed deep skepticism with Kerry's plan, saying Russian air strikes have primarily targeted U.S-supported moderate opposition groups.

"How serious are they (Russia) about defeating terrorism, as opposed to bolstering Assad’s position at the center? How are we to assume that they are going to do better if they work with the Americans?" Ford told Reuters.

Ford said that Russia has been unable to get its "obstreperous client" Assad to make any political concessions, allow significant humanitarian assistance into numerous rebel-held enclaves or abide by a cessation of hostilities agreement.

"I don't think the Russians can deliver," he said. "I don't see how this American-Russian condominium on the Islamic State can help bring about a negotiated solution to the broader Syrian conflict. It will make it less likely that Assad will make concessions that will bring the broader civil war to an end."

(Additional reporting by John Walcott and David Alexander)