He has missed the filing deadlines in two states, Alabama and Arkansas, so he begins at a disadvantage should the primary evolve into a marathon where every delegate is crucial. And he’s already started to draw fire from liberal critics for his current post at Bain Capital, the private equity firm that Senator Mitt Romney of Utah co-founded and led, and that Democrats assailed Mr. Romney for when he ran for president in 2012.

There is also the more fundamental question of whether there’s even an opening for a new candidate. Polls of Democratic voters indicate that they’re mostly satisfied with what’s still an unusually large field of contenders.

Yet while even Mr. Patrick’s most ardent admirers allow that he faces long odds, they believe that the splintered nature of the race calls out for an upbeat and consensus-oriented candidate.

“If anybody is capable of catching lightning in a bottle, it’s him,” said Tim Murray, a former mayor of Worcester, Mass., who served as Mr. Patrick’s lieutenant governor.