Most sports fans in England, Wales and Scotland say they would be comfortable with their club signing a gay player, according to a BBC Radio 5 live survey.
It found 82% of supporters would have no issue with a gay player.
However, 8% of football fans said they would stop watching their team.
Last week, Football Association chairman Greg Clarke told MPs he was "cautious" of encouraging a player to come out because they may suffer "significant abuse" from fans.
Speaking to BBC Radio 5 live on Wednesday, Clarke said he stood by his "personal view" that "vile abuse" from a "small minority on the terraces" must be solved before any gay footballers "take that risk" to come out.
"If they want to take that risk I would respect them and support them," said Clarke. "But we can't promise to provide them at the moment with the required protection. We need to redouble our efforts to provide that safe space."
Clarke added that he hoped to achieve that in a "year or two".
In an online survey of more than 4,000 people - 2,896 of whom were sports fans - commissioned by Afternoon Edition and carried out by ComRes, 71% of football fans said clubs should do more to educate fans about homophobia.
And 47% of all sports fans - 50% of football supporters - say they have heard homophobic abuse at matches.
Former Premier League striker Chris Sutton told Afternoon Edition that Clarke had "taken the easy way out" by being "dictated to by 8% of cavemen".
Sutton, who played for Norwich, Blackburn, Chelsea and Celtic, said: "Coming out wouldn't be a problem in the workplace. Working at a football club is just like anywhere else. Players I played with wouldn't bat an eyelid.
"This 8% shouldn't be allowed in football grounds. By not taking it on, the 8% are the winners in all of this. Greg Clarke should be taking these people on.
"It's bonkers in our society that people like this can dictate whether someone can come out or not."
Simone Pound, head of equality and diversity at the Professional Footballers' Association, told BBC Sport the PFA and the FA were not "blaming any one particular group" for a lack of visibly out gay players.
"I have worked in the game for over 15 years and I have certainly seen a shift in the culture as well as greater understanding and acceptance of LGBT people," she said.
"Coming out is a personal journey that is up to each and every individual. The PFA will continue our work tackling homophobia until someone does come out and thereafter."
An FA spokesman said it "welcomed the statistics" as a "sense check" on homophobia. It said it takes "strong action" against anyone found guilty of "homophobic, biphobic or transphobic abuse".
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What else did the survey say?
On Wednesday, gay rugby league player Keegan Hirst, 28, guest edits a special Afternoon Edition programme on homophobia in sport on BBC Radio 5 live from 13:00 BST.
The survey of fans of 11 different sports also found:
- More sports fans (12%) would feel uncomfortable with a rival player joining their club than a gay player (8%)
- 7% of sports fans would stop watching their team if they signed a gay player
- 57% of sports fans believe gay players should come out to help others do the same
- 18% of sports fans believe gay players should "keep it to themselves"
- 15% of sports fans think having a gay player on a team would make other team-mates feel uncomfortable
- 50% of football fans say they have heard homophobic abuse, 51% have heard sexist abuse and 59% have heard racist abuse
Are there any gay footballers?
Justin Fashanu became the first player in England to come out as gay in 1990, but took his own life in 1998, aged 37. No male professional player has since come out while playing in England.
Former Germany and Aston Villa player Thomas Hitzlsperger became the first player with Premier League experience to publicly reveal his homosexuality, in January 2014, after he had finished playing in England.
American ex-Leeds United winger Robbie Rogers announced his retirement at the same time as revealing he was gay, saying it was "impossible" to come out and remain in the game, although he subsequently returned to football with American team LA Galaxy.
Former England women's captain Casey Stoney was the first active footballer to come out in England since Fashanu, in February 2014.
Swedish lower league player Anton Hysen, son of former Liverpool player Glenn Hysen, publicly announced his homosexuality in an interview with a Swedish football magazine in 2011.
How does the UK compare globally?
Out On The Fields, a two-year global study of homophobia in sport, examined the experience for players and spectators in English-speaking countries including the UK, United States, Canada and Australia.
Study manager Erik Denison said "rates of homophobia in sport were rampant right around the world".
He told BBC Radio 5 live that the UK had "many more" young people willing to come out to their team-mates than in other countries, but that LGBT spectators felt they were "not very safe" at sporting events in comparison.
He said: "What was a bit alarming in the UK was that young people were more likely than older generations to say that they had personally been targeted by homophobia in the form of slurs, bullying, and assaults."
My tweet was 'totally wrong'
Coventry City defender Chris Stokes was banned, fined and sent on an FA education course after tweeting the word "faggots" when commenting on a Chelsea-Tottenham Premier League match in May.
He apologised, immediately removed the tweet and told BBC Radio 5 live that "what I said was totally wrong".
The 25-year-old added: "Nowadays, how the world is, it is a great time for someone who is gay to come out.
"They would get the full support from the changing room - and hopefully the full support of people in the stand."
Listen to Keegan Hirst on Afternoon Edition from 13:00 BST on Wednesday, 26 October.