Mr. Brown and Ms. Schultz noted that they were hearing from a broad range of people but declined to offer any names.
He said he was staying out of the race and had no regrets. The Ohio senator said he was confident Democrats would eventually rally behind their nominee, but he warned the party not to embrace a single-payer health care plan that eliminates private insurance.
“I think it’ll be a hard sell to the public if we go into the general election for ‘Medicare for all,’” said Mr. Brown, citing the risk of alienating union workers who would lose their negotiated plans.
One longtime Democrat who originally sought to entice Mr. Brown into the race, Randi Weingarten, the president of the American Federation of Teachers, could not help letting out a loud “oy!” when asked about the possibility of another candidate joining the sprawling field.
“For as long as I have been in politics, I’ve heard Democrats fretting about their presidential contenders,” said Ms. Weingarten.
Indeed, for some Democrats, the grass is always greener outside their field.
There were multiple stages of the 1992 primary when Bill Clinton’s candidacy was seen as doomed, either because of his own vulnerabilities or because of the third-party threat of H. Ross Perot, the wealthy Reform Party candidate. Would-be Democratic saviors that year included Gov. Mario Cuomo of New York, former Senators Lloyd Bentsen of Texas, Sam Nunn of Georgia and Al Gore of Tennessee, and Representative Richard Gephardt of Missouri.
In the 2004 race, Mr. Gore was again sought after as a potential candidate. That race evolved along similar lines to the current primary, with Democrats desperate to oust an incumbent Republican (George W. Bush) but nervous that their front-runner into the fall (Gov. Howard Dean of Vermont) would prove unelectable as the nominee. That time, a candidate did come in relatively late in the race, Wesley K. Clark, a retired general, but he gained little traction and Mr. Kerry ultimately won the nomination.