Rodgers joined the English Premier League side on Tuesday with Neil Lennon returning to Celtic on an interim basis.
The Hoops hold an eight-point lead in the Ladbrokes Premiership title race and remain on course for a treble treble.
Despite this, Durham claims Rodgers was afraid of his legacy at Parkhead being tarnished by the mounting challenge from Gerrard.
He said: “I think he’s actually scared of Steven Gerrard.
“The fact that Rangers beat Celtic just after Christmas, that’s a worrying sign for Brendan Rodgers.
“We all know Celtic will win the title but if you look at how much they’ve won the title by that’s the key.
“The gap with Rangers is getting closer and closer. He doesn’t want to be the guy who’s in charge when Rangers overtake Celtic.
“It’s a little bit about bottle and a bit about managing your own career but I think Gerrard has got him running scared a little bit and that’s why he’s coming to Leicester.”