With Hodgson ruling out a return for John Terry to the international setup, Begovic believes the coach should give Shawcross the chance to impress.
England have plenty of options at centre-half, with Gary Cahill, Phil Jagielka, Chris Smalling, Phil Jones and Joleon Lescott all used in recent matches ahead of Shawcross.
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"I've been beating the drum for Ryan for a while," he told the Stoke Sentinel. "I get to train with him and play with him every week and see what kind of quality player he is.
"I don't think he would ever let England down.
"It's important to have leaders and he's been our captain for a number of years. It's someone they need to look at and make a decision.
"There's no doubt you do get overlooked if you're not playing for one of the (title-chasing) clubs.
"I wish he was Bosnian! He's a great player, and I'm sure he will get his international chance soon."
Shawcross' solitary England cap came in a 4-2 friendly defeat against Sweden in November 2012 and he has not been part of a squad since.