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The story may be apocryphal. Bill English is also understood to have put his hand up. But he agreed Mr Joyce was the obvious candidate for a task which must be completed as quickly as possible for pressing political reasons yet which cannot be rushed because of its hideous complexity.
Mr Fix-It; the Minder; the minister for everything - you can choose whichever hackneyed label you wish. But behind the cliches lies one incontrovertible fact: all prime ministers need someone like Mr Joyce on hand, especially given that the range, scale and intractability of problems all seem to intensify as a government ages.
Such troubleshooters are the most trusted of lieutenants. They are the safe pair of hands in which a prime minister can place absolute confidence that the job will be done while everyone else is washing their hands of it.
They are jacks of most political trades and masters of the remainder. They can absorb the detail without losing the big picture.
Crucially, they enjoy a level of political seniority which allows them to bang heads together (metaphorically speaking) to make things happen.
Their reward is invisible, yet palpable - ever-increasing influence as the power behind the throne.
Jim Bolger had Bill Birch, someone whose political acumen, understanding of the bureaucracy and relentless work ethic meant he could sort out most foul-ups thrown in his direction.
Helen Clark had Michael Cullen, whose considerable intellect was tapped to do the almost-impossible - come up with a lasting solution to the foreshore and seabed imbroglio.
Mr Joyce has likewise been handed the hospital pass from hell. The problem had become bigger than National had earlier anticipated. Junior minister Craig Foss was struggling to stem the never-ending flow of errors in the payroll.
You only have to read the catalogue of woes in the Ministry of Education briefing paper written for Mr Joyce to see how daunting the challenge is.
The document gets to the nub of the problem - that Novopay was designed with the assumption that 90% of school administrators would be utilising it online, not the little over 70% in actuality.
Worse, some of aspects of the payroll - such as submitting timesheets for part-time teachers - could not be done online.
The result was schools submitted pay forms by email. Talent2, the Australian-based company which owns Novopay, had not devoted enough staffing to cope with this deluge. Exiting staff were not trained to answer the schools' queries.
The ministry said Talent2's performance had been improving, but this was so slow it was virtually invisible to schools, which had largely lost confidence in the new system.
If nothing was done, it would take 18 to 24 months before the system was delivering at the level expected of it let alone generating the extra benefits promised of it.
Not surprisingly, Mr Joyce has recoiled in horror at such a timetable. He certainly warned that there was no quick fix. But that was much to avoid unduly raising expectations among teachers and parents that his appointment would mean rapid progress in sorting out the mess. Mr Joyce would rather adopt the standard approach that it is better to underpromise and overdeliver.
At the same time, he cannot be sanguine about time. The pressure is on him to institute measures which mean a substantial reduction in errors and which have the double plus of taking the pressure off waiting times for teachers seeking answers from Novopay's and the ministry's call centres.
Mr Joyce's inclination is to stick with Novopay simply because things are so far down the track that it would be even more disruptive to switch to a back-up provider.
But he has to be absolutely sure Novopay can be fixed. That is the priority. He has accordingly instituted a technical audit to find out whether Novopay can be ''stabilised'' or whether the project will have to be ditched. That is expected to take three to four weeks.
Until then, he will not foreclose on any options, such as discussing contingency plans with Datacom, the previous supplier.
Among a host of other initiatives designed to be inclusive of sector groups, Mr Joyce has instituted a separate ministerial inquiry, which will assess all aspects of the Novopay project from go to woe.
That will take in the at-times fractious dialogue between the ministry and Talent2, which saw the project's ''go live'' date twice delayed and the ministry threatening breach of contract. Given problems were apparent at least three years ago, one of the more delicate questions will be how much ministers knew and how much the ministry kept them in the dark.
One plus for Mr Joyce and National is that Labour is implicated in this sorry saga by virtue of having signed the contract with Talent2 back in 2008.
But that does not make Mr Joyce's job any the easier. The ministry is already pouring additional funding into the project to try to get it back on the rails. There will likely have to be more.
Mr Joyce hardly needs telling that there is not a lot of political upside in New Zealand taxpayers having to bail out an Australian enterprise which has displayed little remorse for its failure to deliver on its contract or the consequent disruption that has caused to many teachers' lives.
But then there is not a lot of upside for Mr Joyce in all of this. Success is simply expected of him; failure will be a major blight on any higher ambitions. So far it has been textbook stuff from the fourth-ranked minister in the Cabinet. In short, so far so good. But there is still a long way to go before he and National can finally breathe easier on this one.
- John Armstrong is the political correspondent for The New Zealand Herald.