Prime Minister Boris Johnson says his Government is working to get crowds back into venues "as soon as possible".
Reports emerged on Tuesday evening that the Government was exploring the possibility of spectators being allowed into venues by Christmas in areas with the lowest coronavirus infection rates.
The PM is reported to have privately told MPs that reopening sports grounds was "a personal priority".
Johnson did not give any specifics on the issue however when the matter was raised during Prime Minister's Questions on Wednesday.
"I understand the frustration over fans and we hope to get crowds back in the ground as soon as possible," he said in answer to a question from Karl McCartney, the Conservative MP for Lincoln.
Former FA chairman David Bernstein, who was at a meeting of football leaders which also involved Culture Secretary Oliver Dowden on Tuesday, told BBC Radio Four's Today programme: "There is an obvious desire and a need to get fans back into stadia but in reality this will depend on the progress with Covid.
"We have to be realistic and I suspect, if it happens, it will be a fairly small number of fans in areas that have a lower problem with the rate of Covid.
"I wouldn't be overly optimistic but clearly everybody wants this is as soon as possible."
McCartney also asked Johnson about an EFL rescue package.
He said: "We don't want any football team to go out of business as a result of this pandemic and we're doing everything we can."
The Government has repeatedly said it expects the Premier League to provide financial assistance to the EFL.
The top flight has made offers of support and clubs in Leagues One and Two are now in negotiations over a £50million rescue package from the Premier League understood to be made up of £20million in grants and £30million in loans.
Clubs and governing bodies across a range of sports have been hard hit by the decision to press pause on the decision to allow a limited number of spectators into venues from October 1 because of rising infection rates.
The second wave of infections has led to a second national lockdown, but reports from Tuesday suggested that if there is a return to a tiered approach to restrictions from early December, then venues in the lowest-risk areas could open up.
The petition to exempt golf courses from the list of venues required to close during lockdown will be debated on November 23.
The All-Party Parliamentary Group for Golf (APPGG) had previously admitted defeat in its bid to keep courses open after writing to Boris Johnson on behalf of England Golf, the PGA, the R&A and other leading groups from the industry.
However, the petition, which has received more than 257,000 signatures, will now be debated in Westminster Hall, along with a similar one relating to gyms.
Catherine McKinnell MP, chair of the petitions committee, will lead the debate. The Department for Digital, Culture, Media and Sport will send a Minister to respond.
England Golf chief executive Jeremy Tomlinson said: "We know the benefits our sport can provide for both mental and physical health at this time of isolation and hope the Government can recognise this during the upcoming debate and support golf's return to playing.
"Earlier this year, the game of golf was able to clearly demonstrate that it could be enjoyed in a Covid-safe way, which is why it was one of the first sports to return once restrictions were lifted.
"This was a privilege that the sport took incredibly seriously and accordingly adopted the highest possible standards to keep participants safe."
