Hart has been in inconsistent form for over a year and a number of recent errors have led to question marks over his position.
His closest rival to the jersey, West Brom's Ben Foster, is currently out injured and Hart will undoubtedly start England's final two World Cup qualifiers against Montenegro and Poland.
Shilton, who made a record 125 appearances for England, admits Hodgson has little choice but to keep faith in the out-of-sorts Manchester City man.
"Joe's brought it on himself a little bit because he has been very inconsistent over the last year and even his own club manager Manuel Pellegrini is starting to make noises about him," the former Leicester keeper told Press Association Sport.
"Whether he's taken his eye off the ball, I don't know. He got in the team and was full of confidence and a breath of fresh air for the England team but at the moment he's put himself under pressure.
"The only way to get out of it is to play, keep clean sheets, roll up your sleeves and train hard and stay in that England team.
"If we win the next two games and keep clean sheets in them then he can build on that. But he's got to get rid of these silly errors because he can't afford them. People will point the finger."
Shilton, running a penalty shoot-out masterclass at the Salvation Army's annual five-a-side tournament for people in homeless centres, added: "Roy Hodgson is in a no-win situation at the moment.
"Ben Foster isn't fit and the other two, John Ruddy and Fraser Forster, have played less than one full game between them and you could have the Scott Carson scenario. He was put in for the crucial Euro 2008 qualifier against Croatia, made an error on the first shot they had and we didn't qualify.
"Roy Hodgson has got to play Joe Hart. He's the most experienced goalkeeper and generally he's done well for England.
"Let's hope he doesn't have a lot of work to do in the two games and he's just a spectator and we do the business at the other end so his form doesn't matter."