WASHINGTON — President Obama’s compromise with Republicans on extending tax cuts for the wealthy, which his self-described progressive critics see as a profound betrayal, is bound to intensify a debate that has been bubbling up on liberal blogs and e-mail lists in recent weeks — whether or not the president who embodied “hope and change” in 2008 should face a primary challenge in 2012.

The idea seems to have little momentum for now, not least because there isn’t an obvious candidate, and because such a challenge would seem to have about as much chance of success as, say, a reality show about David Hasselhoff. That a primary is being openly discussed, though, reflects how fully Mr. Obama’s relationship with his party’s liberal activists has ruptured and the considerable confusion on the left over what to do about it.

Just last weekend, three liberal writers made the case for taking on Mr. Obama in 2012. Michael Lerner, longtime editor of Tikkun magazine, argued in The Washington Post that a primary represented a “real way to save the Obama presidency,” by forcing Mr. Obama to move leftward. Robert Kuttner, co-founder of The American Prospect and one of the party’s most scathing populist voices, issued a similar call on The Huffington Post, suggesting Iowa as the ideal incubator.

On the same site, Clarence B. Jones, a one-time confidant of the Rev. Dr. Martin Luther King Jr., suggested that liberals should break with Mr. Obama now, just as Dr. King and others did with Lyndon B. Johnson in 1968. “It is not easy to consider challenging the first African-American to be elected president of the United States,” Mr. Jones wrote. “But, regrettably, I believe the time has come to do this.”