The Englishman has led the club to the Betfred Cup semi-final and a draw against Celtic in the past week, strengthening his position in the eyes of the bookies.
Hibs were impressive on Saturday as they became the first side to take points off of the Scottish champions this season.
Aberdeen manager Derek McInnes has replaced him at 7/4 as the first manager to leave his post in the Ladbrokes Premiership.
The Dons manager has had his odds slashed from 6/1 McBookie following the side's 5-0 defeat to Rangers on Saturday.
McInnes watched on as his side were thumped by Steven Gerrard's rampant team at Ibrox.
He said: "Aberdeen, with our experienced, battle-hardened players, can still win these games. There is no question of that - but we looked miles away from that.
"We looked like a team playing our own wee game and we lacked the personality and experience you need to get a result at Ibrox."
Heckingbottom's price has drifted to 11/4, making him second favourite for the chop. However, there have been encouraging signs over the last seven days to suggest that price could get longer.
Hibs, of course, travel to face Aberdeen on Saturday.
Meanwhile, Craig Levein's price has also drifted. Now the Heart boss sits at 6/1 after the goalless draw at St Mirren over the weekend.
Odds on the 12 Scottish Premiership managers to be the first to leave their post:
Derek McInnes (Aberdeen) - 7/4
Paul Heckingbottom (Hibernian) - 11/4
Craig Levein - 6/1
Tommy Wright (St Johnstone) - 8/1
Angelo Alessio (Kilmarnock) - 14/1
Neil Lennon - 14/1
Brian Rice (Hamilton Academical) - 16/1
Gary Holt (Livingston) - 20/1
Jim Goodwin (St Mirren) - 20/1
Stuart Kettlewell/Steven Ferguson (Ross County) - 20/1
Stephen Robinson (Motherwell) - 33/1
Steven Gerrard (Rangers) - 33/1