WASHINGTON (Reuters) - To Republican U.S. presidential contenders, North Korea’s claim that it tested a hydrogen bomb may further make the 2016 race what they dearly want it to be: a referendum on President Barack Obama's foreign policy and, by extension, Hillary Clinton’s.

For months, these Republicans have liked to say the world is "on fire," pinning the conflicts in Iraq and Syria, the attacks in Paris and San Bernardino, California, and the recent tensions between Iran and Saudi Arabia on Obama’s administration and Clinton’s stint as his secretary of state from 2009 to 2013.

Now, they can add North Korea to the threats they say face American voters.

"When China fell to the communists (in 1949), the question that dogged the Truman administration was: 'Who lost China?'" said John Feehery, a Republican strategist. "The question that will dog the Democrats is: Who lost North Korea?"

The criticism on foreign policy has ratcheted up the pressure on Clinton, the likely Democratic presidential nominee in November's election, to take a harder line on national security without handing Republicans more ammunition to argue that Obama's stewardship has been a failure.

Analysts said Republicans may have little room to maneuver since the Obama administration's approach toward containing North Korea did not differ materially from the one used by Republican George W. Bush's administration before it.

"They’ve been a headache for every Democrat. They’ve been a headache for every Republican," Michael Rubin, a scholar at the conservative American Enterprise Institute, said of the North Koreans. "North Korea may be the last remaining foreign policy quagmire that hasn't been politicized in a partisan fashion."

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