The Celtic boss is now joint-third favourite along with ex-Chelsea manager Antonio Conte at 10/1. He previously sat as a 66/1 outsider.
Zinedine Zidane and Mauricio Pochettino lead the way in the eyes of the bookmakers.
Pressure is increasingly on United boss Jose Mourinho after the Red Devils made it four games without victory last night as Valencia held the hosts to a 0-0 draw at Old Trafford.
Rodgers has previously reaffirmed his commitment to Celtic amid supposed interest from England.
The Celtic favourite also revealed that an unnamed Chinese club had enquired as to his availability earlier this year.