I read somewhere that John was a Liverpool F.C. fan and that he even dreamt about becoming a professional footballer but soon realized that he isn't talented for it. It was Lennon's idea to put Albert Stubbins, a legend of Liverpool FC on the cover of Sergeant Pepper. Macca's supporting was discussed several times already. His father was supporting Everton F.C. (also a club in Liverpool), but he said that he supports both Everton and Liverpool. But if it comes to a derby between the 2 teams he supports Everton.
Ringo was a fan of Arsenal (a club in London). I think he was actually registered as an official fan. I don't know about George though.
Overall I believe they weren't huge football fans.
Some sources to make it more convincing
"Here's the deal: my father was born in Everton, my family are officially Evertonians, so if it comes down to a derby match or an FA Cup final between the two, I would have to support Everton.
"But after a concert at Wembley Arena I got a bit of a friendship with Kenny Dalglish, who had been to the gig and I thought 'You know what? I am just going to support them both because it's all Liverpool and I don't have that Catholic-Protestant thing.
"So I did have to get special dispensation from the Pope to do this but that's it, too bad. I support them both.
"They are both great teams.
"But if it comes to the crunch, I'm Evertonian." Macca
http://www.evertonbanter.co.uk/2008/07/sir-paul-mccartneys-everton-se.htmlAs a proud support of Manchester United football club I have to add the lyrics of Dig It.
Like a rolling stone
A like a rolling stone
Like the FBI and the CIA
And the BBC--BB King
And Doris Day
Matt Busby
Dig it, dig it, dig it
Matt Busby is a legend, a former manager of Man United (Liverpool's biggest rivals). The song Dig it is featured on Let It Be and credited to the whole Fab 4. Although Busby if one of the biggest legends Man United he was much more than this. In the Sixties he was a major icon of British football which is probably the reason why he is mentioned in the song. It has nothing to do with the Beatles supporting Man United.
*Oh and that picture that you posted is believed to be the 52 FA Cup Final between Arsenal and Newcastle. (The former won 1:0)