WASHINGTON — The Democrats’ widespread losses last week have revived a debate inside the party about its fundamental identity, a long-running feud between center and left that has taken on new urgency in the aftermath of a disastrous election and in a time of deeply felt economic anxiety.

The discussion is taking place in postelection meetings, conference calls and dueling memos from liberals and moderates. But it will soon grow louder, shaping the actions of congressional Democrats in President Obama’s final two years and, more notably, defining the party’s presidential primaries in 2016.

“The debate will ultimately play out in a battle for the soul of the Clinton campaign,” said Matt Bennett, a senior official at Third Way, the centrist political group.

Hillary Rodham Clinton, should she run, will face tension between the business-friendly wing of the party, which was ascendant in the economic boom during her husband’s administration, and the populism of Senator Elizabeth Warren of Massachusetts, which has gained currency of late.