WASHINGTON — As President Obama made the case on Wednesday about how the American economy was improving, he went off script to underscore how significant the challenges were that remained.

“But — and here’s the big but — I’m here today to tell you that we’re not there yet,” Mr. Obama said, expanding on his prepared text to note that the wealthy are doing better while the middle class is struggling and others are faring worse.

He did not need to remind many of his fellow Democrats.

With Mr. Obama experiencing a difficult first year of his second term and his lame-duck status growing ever nearer, his speech underscored the stirrings of a debate inside the Democratic Party about the party’s economic approach, given the halting recovery.

If positions on foreign policy and specifically the Iraq war marked the dividing line in the Democrats’ last fierce internal debate, issues related to banks, entitlements and the rights of consumers broadly could shape the party’s next search for an identity.