In the leafy garden of a laid-back Italian restaurant next to Fifa headquarters, four men sit at a table, talking earnestly, even tensely.

"We just have to keep on, keep on going," one says.

These are Fifa men, and one of them is Walter de Gregorio, Fifa's director of communications, and the man who had to sit alone on Fifa's vast auditorium stage last Wednesday morning and explain to the world's press just why it was that seven senior football officials had been hauled from their beds by Swiss police early that morning, and detained pending extradition.

In the six days since, events concerning Fifa have moved with astonishing speed.

Its controversial president Sepp Blatter was asked to resign, instead he stood again and was re-elected with a handsome number of votes.

More allegations of bribery and corruption followed, and within days Mr Blatter had resigned after all.

Start at the top

So what, I asked Mr de Gregorio, did he think it would take now to fix Fifa?

Reform must begin with the executive committee, he indicated.

"It's no use the president saying he can't monitor everyone all the time," he said, in reference to Mr Blatter, who said exactly this just last week when bidding for re-election.

Mr de Gregorio suggested that unless there was more oversight of Fifa's executive committee members, then whoever became the new president would have the same problems as Sepp Blatter.

So a new structure built to prevent quiet deals (some would call them bribes) and, Mr de Gregorio said, obligatory "integrity checks".

The integrity of integrity checks

But what exactly is an integrity check? The anti-corruption organization Transparency International (TI) has been calling for such checks for quite some time.

"But," says Gareth Sweeney, who is director of TI's corruption in sport initiative, "it depends on the integrity of the integrity check."

To be really effective, Mr Sweeney says, there would have to be rigorous background checks - not just for criminal records, but about previous employers and current professional contacts, in order to ensure there is absolutely no conflict of interest.

Root-and-branch reform

TI would like to see fixed terms for everyone on Fifa's powerful executive and finance committees, as well, of course, for its president. So no more decades in office for Fifa bosses.

Another suggested change would be to hold open rather than secret ballots.

This, some suggest, would force national football associations to be accountable. Instead of going into a booth and casting a secret ballot, as 133 did last week for Sepp Blatter, the associations would have to vote publicly.

Finally, TI believes Fifa should have external independent directors on its executive and finance committees too, in a similar way to the International Olympic Committee, which, in the hope of repairing its reputation after suffering its own corruption scandals, now has more checks and balances than are actually legally required.

But in Switzerland itself, some Fifa watchers believe the damage of the last week may not be repairable.

Guido Tognoni, a former marketing manager for Fifa, believes the organisation is heading for "destruction", and that tinkering with its current set-up is not an option.

Dismantle Fifa?

A Swiss Social Democrat member of parliament, Daniel Jositsch, has suggested a complete dismantling of Fifa as it stands now, hiving off its commercial side into a limited company, complete with shares and shareholders, and governed by business law.

The purely sport side would become a not-for-profit foundation.

Perhaps surprisingly, another member of the Swiss parliament and one of Fifa's fiercest critics, Roland Buechel, is less pessimistic.

He says he is prepared to be "positively surprised" by Sepp Blatter, who, having announced his resignation and now with just a few months left in office, may finally achieve serious and effective reform.

To do so he will have to commit to it though, something he has not really done in the past. And he will have to take on football's powerful confederations, the regional groups which national football associations join.

It is these confederations which have resisted change in the past, and which may resist again.

Gareth Sweeney of TI fears the groups who voted to keep Sepp Blatter in power may now mobilise around a new presidential candidate who is reluctant to tackle reform.

All agree though that the fate of those seven senior football officials who were unceremoniously removed from their five-star Zurich hotel last week and taken to a Swiss jail may be the best catalyst for reform.

Apparently, the seven gentlemen, while resisting extradition to the United States, are also rather unhappy with the very basic standards of their Swiss prison cells.

And there are those two criminal investigations. However many reforms Fifa introduces, the investigations are under way and will continue.

"Reputationally," says Gareth Sweeney, "it's very likely things are going to get a lot worse for Fifa before they get better."