Not necessarily in helping to run them, though this is a distinct possibility, but in ensuring that the event does not take place without a Great Britain team in the football tournament.
He says: "It is unthinkable that we do not compete as an all-British team with representatives from all the home nations. In my opinion we owe it to the generation of young men and women, able-bodied and disabled, to take part in the Olympic and Paralympic football tournament, in matches, by the way, that do not just take place in London but all over the country, and Scotland and Wales too.
"I wouldn't say that it has become an obsession, but finding a solution has become a keen priority for me before I leave the FA."
Among his varied duties as the FA's man for all seasons, Davies, as executive director, has been football's representative on the British Olympic Association since 1999. Before then the chair was all too frequently left empty, but Davies says: "I believe there is a lot football can learn from other sports.
"I never really understood the Olympic football issue until I had responsibility for it. I studied its history, and the political concerns of the Celtic nations are nothing new - at one stage the Scots insisted that they had four players in the squad and four in the team.
"But I consider it inconceivable that when the Soviet Union has gone in 15 directions and Yugoslavia in four, and there are Scottish and Welsh parliaments, that anyone should seriously question the inviolacy of those nations.
"Sepp Blatter has given them absolute assurance that their independence will not be affected, but we have to come up with proposals that are attractive enough to convince them."
Davies's enthusiasm for the Olympics, allied to his connections in Westminster and Whitehall, dating from his days as a BBC lobby correspondent, are unlikely to be wasted in another life. He says he has no definite plans for his future at 57, but a significant contribution to 2012 would be no big surprise.
He insists that his decision to quit after what will have been a 13-year rollercoaster ride with the FA was made before the appointment of the fellow ex-media man Brian Barwick - the fourth chief executive he has worked with at the association - and also before the scandal involving his former PA, Faria Alam, and her involvement with Eriksson and Barwick's predecessor, Mark Palios.
Her subsequent allegations of sexual harassment against Davies himself were thrown out by an industrial relations tribunal, but it is apparent that he was deeply upset by them, though it was by no means the only storm weathered in a career which saw him as the man on the doorstep of the FA fronting out crises from bungs to bungling, hooliganism to Hoddle. A long way from the time he presented Songs of Praise on television and was ribbed as "The Bishop".
"Some of those times have been pretty awful," he said. "Especially recently, but the good times have easily outweighed the bad. I don't spend my time moaning about my former profession because I find that counter-productive. I suppose the coarseness of the criticism is a product of the times we live in, but I have got a bit tired of some of the personal insults that fly around. You do have to be thick-skinned, and unfortunately I'm not."
Davies, the great survivor, has seen out four coaches as well as four chief executives, and spent two lengthy periods as stand-in chief executive, a job he said he never coveted.
"We have come through a tricky period. No one pretends that we have played well since August until Wednesday night. The trouble is that there is very little middle ground in sport these days. You are either wonderful or absolutely useless. We are not either.
"We are promising, one of a number of teams who could win the World Cup. They certainly won't be queuing up to be in England's group in Germany. We have outstanding players, probably the highest number in my time here. But you need some good fortune too.
"I believe that football does benefit from the success of other sports. It did after England won the Rugby World Cup, and certainly after the Ashes this summer. I believe it will now benefit enormously from the Olympic bid. But boy, imagine what the reaction would be if we did win the World Cup!"
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