The Sun's "Bwing on the Euwos!" front-page headline today - on the appointment of Hodgson on a four-year contract - has attracted "a large number of objections" to the FA and the Press Complaints Commission (PCC), said the football body.
The story added: "(We'll see you in Ukwaine against Fwance)." The previous day, the paper had also run a line on its front page describing Hodgson as 'Woy', rather than Roy.
The FA said that it would not make an official complaint to the PCC, but it has raised the issue with The Sun and "made it clear that their front page is unacceptable to us".
"The FA and the Press Complaints Commission have today received a large number of objections relating to the front page headline in the Sun newspaper, regarding Roy Hodgson's manner of speech," said the FA in a short statement.
"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."
FA chairman David Bernstein added: "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."
The Sun, owned by Rupert Murdoch's News International publisher, responded by saying that West Bromwich Albion boss Hodgson was "affectionately known as Woy due to his speech impediment".
Watch Roy Hodgson being unveiled as the new England football manager below: