The 64-year-old was confirmed in the role on Tuesday, and The Sun newspaper mocked his slight speech impediment in a front-page story the following day, while other newspapers reported erroneously that Germans had never heard of him.
The English FA responded with an immediate complaint. This all happened a month before Hodgson even takes charge of England for the first time with pre-Euro 2012 friendlies against Norway and Belgium.
Asked by reporters if he had a thick skin, Hodgson replied: "Maybe I should develop one.
"I'm a football coach. That's been the whole of my life. Dealing with the mass media has been a part of that. I have not shied away from it and it hasn't bothered me. It's part of my duties. I get on with it and do a reasonable job.
"But my forte is coaching footballers and preparing, building and improving football teams. That's basically what the England manager's job is.
"If I'm going to be vulnerable in any area it might be that I don't have a thick enough skin. But I'd rather that than not have the wherewithal to do the job."
Part of the media's problem with Hodgson's appointment is that they had championed Tottenham manager Harry Redknapp for the role.
Hodgson, who has also managed the national teams of Switzerland, Finland and the United Arab Emirates, said he did not think for a second of turning down the FA's offer.
"There was no element of doubt," he said. "I have been a candidate in the past and it's gone to other people. When this opportunity came around, even though I maybe should have considered these things, I haven't.
"I'm just delighted and pleased to have the opportunity to lead my country and help the team get success."
Hodgson said he would name his 23-man squad for the Euros later this month and might seek advice from his predecessor Fabio Capello.
"It's a good idea," said Hodgson. "No doubt I will try and speak to him, if I can track him down, although it's not something I've had a chance to do yet."