England players took to Twitter on Thursday to defend their manager, who was also backed by the English Football Association, but anti-racism campaigners are calling for an investigation into an incident that has taken some of the shine off England's qualification for next year's tournament in Brazil.
Hodgson has apologized after recounting an old joke about space monkeys in the England dressing room during Tuesday's match against Poland, as he attempted to explain how he wanted his players to pass the ball more to winger Andros Townsend.
Townsend is black and the word "monkey" can have racist connotations.
"I would like to apologize if any offence has been caused by what I said at halftime," Hodgson said in a statement issued late Wednesday. "There was absolutely no intention on my part to say anything inappropriate. I made this clear straightaway to Andros in the dressing room.
"I also spoke to Andros again on Wednesday. He has assured me and the FA he did not take any offence, and understood the point I was making in the manner I intended."
Townsend backed up Hodgson's comments by tweeting his reaction to a controversy that was getting plenty of play on social media sites.
"I don't know what all this fuss is about," the Tottenham winger wrote. "No offence was meant and none was taken! It's not even news worthy!"
England striker Wayne Rooney added on Twitter that the story was "ridiculous" and that Hodgson has "done nothing wrong."
Later Thursday, FA chairman Greg Dyke defended Hodgson's character and said the organization has talked "extensively" to squad members about the incident.
"Roy Hodgson is a man of the highest integrity, an honourable man who is doing a great job with the England team. He has and deserves the full support of The Football Association," Dyke said.
"He has made clear there was no intent to say anything inappropriate," Dyke added, "and he was certainly not making any comments with any racist connotation."
However, anti-racism group Kick It Out is demanding an investigation "to ascertain the full facts and ensure a similar situation does not arise again."
"Kick It Out ... recognizes and shares the concerns of the parties who felt mindful to bring the comments into the public domain," the group said in a statement.
Piara Powar, executive director of European anti-discrimination group FARE, said on Twitter that Hodgson had "used a very silly term within a diverse team environment. He should know better."
There was also mystery regarding who leaked details of Hodgson's team talk to The Sun newspaper, which published the story on its front page on Thursday.
"Love how people blaming me already, and i wasn't even there, anything bad in the world #blameash ," tweeted Ashley Cole, who didn't play on Tuesday because of injury.
Hodgson is reported to have used a joke about space monkeys that became popular at NASA in the 1960s and '70s, after the American space agency sent monkeys into space before humans.
One version sees an astronaut radio NASA to ask what tasks he can do during a mission, with NASA replying: "In 15 minutes — feed the monkey."
"The FA has not had a complaint from any squad member or player representative, and we have today talked extensively to the squad," Dyke said in the FA statement. "The FA has been assured by the players that there are no problems and they understand the point Roy was making and the context in which he was speaking."
England beat Poland 2-0 at Wembley Stadium, sealing qualification for a fifth straight World Cup.
The FA announced Thursday that England will play fellow qualifier Chile in a friendly at Wembley on Nov. 15. Hodgson's side hosts Germany at the same stadium four days later.