The Sun's headline 'Bwing on the Euwos!' has led to numerous complaints to the FA and the Press Complaints Commission, an FA statement said.
FA chairman David Bernstein said they had been delighted with the media coverage generally but that the headline was "disrespectful".
The statement said: "The FA and the Press Complaints Commission have received a large number of objections relating to the front page headline in The Sun newspaper, regarding Roy Hodgson's manner of speech.
"On this occasion, we will not be making an official complaint to the PCC but we have raised it with the newspaper and made it clear that their front page is unacceptable to us."
Bernstein said: "We are delighted at the media response to Roy's appointment but are disappointed with the headline in The Sun, which we consider is in poor taste and disrespectful."
In its story the Sun said Hodgson was "affectionately known as Woy due to his speech impediment".
A spokesman for the Press Complaints Commission (PCC) said: "I can confirm we have had over a hundred complaints."
News International has yet to make any comment on the headline.