The Football Association has thrown its full support behind Roy Hodgson after speaking directly to every member of the national squad involved in Tuesday's World Cup qualifying victory over Poland and ascertaining that none had taken offence at the punchline to a "space monkey" joke Hodgson had relayed at half-time.

The England manager, meanwhile, was left furious that the feelgood factor generated by the qualification for the World Cup had been immediately blown away by the furore. "The players are as angry about this as I am," he told the Daily Mail. "We have just had a successful period and, although I wouldn't suggest we intend to rest on our laurels, I think we have earned the right to enjoy the fruits of our labours. Instead we get this..."

Hodgson had used the phrase "feed the monkey" during the interval at Wembley in an attempt to encourage his players to give the ball more often to the winger, Andros Townsend. It is unclear whether the manager relayed the whole of what he clearly considers to be a well-known joke, which became popular at the US space agency, Nasa, in the 1960s and 1970s after it sent monkeys into space before humans.

Townsend has since taken to Twitter to stress no offence was taken, though the word "monkey" can certainly have racist connotations. The anti-racism campaign Kick It Out had asked the FA to investigate after the story emerged in Thursday's Sun newspaper.

"You would have to go a long way to find somebody who is less racist than me," said Hodgson. I have coached almost everywhere in the last 37 years, worked with all kinds of people, so this intimation of racism – you can hardly call it a story – is saddening."

The FA's investigation was swiftly undertaken with every member of the 22-man squad present at Wembley for the Poland game consulted in strict confidence on the telephone– if players could not be reached earlier in the day, contact was eventually established with them all – and each given the opportunity to indicate to the governing body that the manager's words had made them uncomfortable. While the FA recognises that the punchline to the rather convoluted joke could easily be deemed inappropriate, it considered the context in which Hodgson was speaking at half-time and duly offered its manager full backing. Hodgson said that he had meant "give the ball to the player who can make things happen. There was no other connotation."

The FA chairman, Greg Dyke said: "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. He has fully explained to us what he said and the point he was making to the players in the dressing room at half-time on Tuesday night. He has also explained the context in which he made his remarks. He has made clear there was no intent to say anything inappropriate and he was certainly not making any comments with any racist connotation. Importantly he has apologised for any unintended offence. that may have been taken. Roy has spoken with Andros Townsend and a number of the players since the game and he has been assured there are no problems within the squad whatsoever.

"Additionally the FA has not had a complaint from any squad member or player representative and we have today talked extensively to the squad. 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. We will be making no further comment on this story and will now be giving Roy and the team our full support as we prepare for the World Cup in Brazil."

That statement appeared to satisfy Kick It Out, football's equality and inclusion campaign which employs Townsend's father, Troy, as its mentoring manager. The chairman, Lord Herman Ouseley, had initially spoken directly to the FA to share "the concerns of the parties who felt mindful to bring the comments into the public domain". They had sought the investigation "to ascertain the full facts and ensure a similar situation does not arise again" and, upon its conclusion, were apparently content with its findings.

"Kick It Out is pleased that the Football Association has investigated this matter swiftly and issued its findings immediately," read a second statement from the body. "If there has been no complaint on the back of the investigation, the matter can only be deemed as concluded.Kick It Out acts on behalf of the football community at large and, when an allegation of a racist or discriminatory nature is made, it's the organisation's role to follow this up."

Townsend, who had excelled on his second appearance for his country on Tuesday, wrote on Twitter earlier in the day: "I don't know what all this fuss is about. No offence was meant and none was taken! It's not even news worthy!" Those sentiments were echoed by his international team-mate, Wayne Rooney, who had tweeted on Thursday morning: "Seen the story on Roy this morning. He done nothing wrong. This is ridiculous."

Rooney later added, via his website: "To be honest it's really annoying that something such as this should see the light of day, all the lads know what type of guy Roy is, and to try and pin some form of label on him is absolutely ridiculous. Roy spoke to Andros straight away who took no offence whatsoever, hopefully that's now the end of the matter."

Yet there was criticism from other quarters, with Piara Powar, executive director of European anti-discrimination group Fare, claiming the England manager "should know better" given he had "used a very silly term within a diverse team environment".

Hodgson, perhaps acknowledging his choice of words might be considered naïve, had felt compelled to offer his apology on Wednesday night in case "any offence has been caused by what I said at half-time as he addressed his players in the dressing room with his team 1-0 up. There was absolutely no intention on my part to say anything inappropriate," he had stated. "I made this clear straight away 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."