Why? Surely it is because Davis is widely thought in Westminster to favour leaving the EU altogether.
It may not be his public position but this does not seem to be a particularly closely guarded secret either. For one example, take Ukip leader Nigel Farage in a Total Politics interview with Davis' former chief of staff Iain Dale, in which Farage spoke about talking to Daniel Hannan during the Tory leadership campaign: "I couldn't believe he would support Cameron. I said we all know what David Davis believes in private so 'why on earth are you backing Cameron?' He said: 'Nigel, because Cameron has made the one deliverable promise [to get out of the EPP]".
(Davis is less popular with veteran Parliamentary colleagues who remember his role in the whips' office in getting Maastricht through, though Davis has since said in interviews that "I didn't want the Treaty" either).
The campaign for a "fundamental renegotiation" is a "false flag" campaign by those - like ConservativeHome editor Tim Montgomerie - who publicly or privately want to get Britain out of the European Union.
Though William Hague comes 4th in the poll of who should conduct Tory renegotiations - where he is the obvious choice, as a Shadow Foreign Secretary with a sceptic reputation - I have been struck by how the party's core Eurosceptics do not put their faith in Hague as their champion in the party's internal debate. There are very frequent references to deputy Mark Francois as sounder on the EU from those most hostile to British membership. (Daniel Hannan describes him as "brilliant ... with a quiet toughness that borders on heroism": high praise indeed).
I first heard the theory that the core Eurosceptics did not regard William Hague as not particularly "sound" on Europe two years ago, during a cordial chat with one of the party's most influential sceptic voices when we were taking part in some media debates together. I was surprised, expressing the view that, were I a Eurosceptic, I would feel that the man who devoted much of his leadership and 2001 election campaign to Eurosceptic campaigning was "one of us".
The intriguing counter-argument was less about the bruising nature of that defeat - which core 'sceptics put down to timing, rather than content - than about the way in which Hague had changed and broadened his non-political interests since 2001.
"With his literary success, William has become part of the Establishment. He is much less interested in rocking the boat".
I don't know how plausible this theory is: Peter Oborne regards Hague as a man "who would at heart like Britain out of the EU altogether", though raising the possibility that he may play more of a Prescott role of selling the leadership to the party than vice versa. But I am confident it is one discussed in the Eurosceptic camp itself.
So who do the core Eurosceptics trust? At that time, David Cameron, who I was told is "much more Eurosceptic than anybody thinks".
But I fear he is about to disappoint the ultras. As Daniel Finkelstein suggests in his defence of Cameron's Euroscepticism
They are properly, robustly Eurosceptic while being completely at home with membership of the EU.
So what Finkelstein calls Cameron's "cufflinks" strategy is a conservative scepticism, which believes it can stem the tide, but is less interested in rewriting the past.
Surely this can never satsify the core Eurosceptics, who are much more revolutionary than that.