Interim England boss Stuart Lancaster would be considered as a suitable candidate for the permanent head coaching role, should he decide to apply.
Lancaster is in temporary charge while Odgers, a firm of headhunters, search for a suitable full-time successor to Martin Johnson, who stepped down at the end of the 2011 World Cup.
Big names like Nick Mallett, Jim Mallinder, Wayne Smith and Rassie Erasmus have all been linked with coaching roles in a new England set-up.
The Rugby Football Union would like to make an appointment before the end of the forthcoming Six Nations, but acting chief executive Stephen Brown insisted that timescale has not been set in stone.
Should Lancaster apply, Brown said he will be given every consideration, with the 42-year-old having already impressed Twickenham executives with his role in the RFU's so-called "reputational damage rebuild plan".
Brown said: "Stuart has made a fantastic impression and I'm sure if Stuart chooses to apply he will be given every consideration for that job.
"Stuart has to make the decision about whether he wants to be considered for the role. If he does then we will make sure there is time (for him to make his case).
"The door is not shut, it's open. All of the good things you have seen Stuart has done have made a massive impact and that is the kind of thing you want.
"It is not just about the results. Clearly that is important and the focus - but we need to build a team that is strong from a performance point of view and from a cultural point of view for the long term.
"They are key bits we need in place and Stuart is focused on those things as well. That is the admirable bit about Stuart (given he is only in temporary charge).
"He is not just focused on winning, he wants to get all of those things in place and you have seen the building blocks starting. To be frank, everything Stuart has done has made a very positive impact."
Lancaster has set about heralding a new era in the England set-up by naming a new-look squad for the Six Nations and taking strong disciplinary action against Danny Care for alcohol-related misdemeanours.
He has also helped the RFU's chief commercial director, Sophie Goldschmidt, take the positive message to sponsors who were concerned about the image of English rugby on and off the field after a turbulent 2011.
"We have worked on a reputational damage rebuild plan since the Rugby World Cup," Goldschmidt said.
"We know how important the (England) team is to everything we are trying to do with the profile they have and we've worked very closely with Stuart Lancaster in particular.
"He has already spoken about some of the cultural areas he is looking to change and improve and that has been tied in very closely with the commercial department in terms of what we want to represent as a brand and what our commercial partners expect from us.
"I think they were surprised by the extent of some of what happened (at the World Cup and afterwards). Clearly they want to see that we are learning from it and putting different measures in place to try to avoid similar incidents happening again.
"To have Stuart leading from the front and championing it has been fantastic. That is definitely having a discernible impact