In one match-up, Patrick leads state Treasurer Tim Cahill, a longtime Democrat running as an independent, by 36 percent to 26 percent with 20 percent for Republican Christy Mihos if he wins his party's nomination. Mihos is a businessman who ran as an independent for governor in 2006.

If the Republican nominee is former health care executive Charlie Baker, Patrick gets 38 percent to 26 percent for Cahill and 15 percent for Baker.

Mihos has a slight edge, 33 percent to 30 percent, over Baker in a Republican primary.