In this June 15, 2015, file photo, demonstrators rally for a $15 minimum wage before a meeting of the state Wage Board in New York.  (AP Photo/Seth Wenig, File)

There are two big splits in U.S. politics right now over the federal minimum wage. One is between Democrats who wish to raise the wage and Republicans who oppose an increase. The other is among Democrats, and it pits President Obama and Hillary Clinton, the party's presidential front-runner, against a growing chorus of liberals.

Obama and Clinton have both expressed support for raising the federal minimum to $12 an hour, up from $7.25 an hour today. Other Democrats want to go further, to $15 an hour, including House Democratic Leader Nancy Pelosi and two of Clinton's opponents for the presidential nomination, Bernie Sanders and Martin O'Malley.

Neither Obama nor Clinton has spoken critically of a $15-an-hour federal minimum. But recent comments from the candidate, and from officials in Obama's administration, suggest Clinton and Obama may be worried that such a large increase could prove too much for some parts of the country to bear.

“We think there needs to be a federal floor" of $12 an hour, "and state and local governments should have the ability to go above that floor,” Labor Secretary Thomas E. Perez said in an interview.

Perez said he has endorsed local efforts to increase the minimum wage to $15 an hour, including showing up in Seattle the day the city's hike went into effect. But he said cost of living should be factored into the national threshold - noting that Seattle is far more expensive than Birmingham, Ala.

In cities such as Birmingham, Perez said, $12 an hour "will enable you to live not like a king but above the poverty line."

Clinton said something nearly identical - with Arkansas subbing in for Alabama - last month in an economic speech, where she endorsed New York's proposal to implement a statewide $15-an-hour minimum for fast food workers.

"The national minimum wage is a floor, and it needs to be raised," she said. "But let's also remember that the cost of living is different in Manhattan than in Little Rock and many other places."

Both Obama and Clinton have raised their target minimum wage hike in recent years. Their hesitation to embrace the so-called "Fight for $15" tracks with concerns voiced in recent months by some left-leaning economists, who warn that such a high federal minimum could force employers to shed a large number of jobs in areas where wages and the cost of living are relatively low - and could hurt low-skilled workers in even higher-cost cities. (Conservative economists tend to warn that even more modest minimum-wage increases would kill jobs.)

"Making 15 the new 10, in terms of minimum wage increases, could potentially generate more harm than good," Harry J. Holzer, a former chief economist at the Labor Department under President Bill Clinton, wrote last month.

If Clinton, in particular, sticks to her support of $12 with regional exceptions, she'll likely be pushed on it in the first Democratic presidential debate in October. Which means she'll have a choice: Simply repeat her position, or challenge Sanders and O'Malley on their support for $15 - which is to say, make the case that the more liberal federal proposal would be a bridge too far.