The RFU are currently dealing with the fall-out after John Steele was ousted as chief executive on Friday following a unanimous vote at an emergency board meeting.
Steele's position was deemed untenable by the RFU board over the handling of the recruitment process for the new performance director role - although Thomas said "other internal aspects relating to dealing with sponsors and stakeholders" had also played a part in the decision.
Since Steele's sacking all eyes have turned to Woodward who is currently the BOA's performance director for London 2012.
Thomas expects to fill the role after the World Cup this autumn and although the Olympics are little more than a year away, he is adamant there would be no way Woodward could fulfil both positions before joining full-time after the Games.
He told BBC Radio 5live's Sportsweek programme: "The England rugby team and its success is critical to me.
"We languish in sixth in the IRB rankings, it's unacceptable.
"I don't believe we can have a situation where a part-timer is doing the job, even if he was to come at a later stage. That for me would not be an answer.
"We need a man that has a skill set that is 100 per cent committed to putting England back to where they need to be.
"We need success in New Zealand and that is totally down to Martin Johnson and his coaches and we then need to be looking forward to 2015, which may sound a long way away but it isn't.
"The RFU job is a full-time job. We have already lost a lot of time."
Of Woodward, Thomas added: "As an Englishman I am a supporter of Clive Woodward to the extent he is the only Englishman walking who has lifted a World Cup.
"He also had a very talented group of players.
"Clive knows my telephone number and I would happily speak to him."
Part of the furore over the recruitment of England's new performance director centred around the manner in which the job description was changed twice in a matter of days last month.
Steele altered the original specification by removing any involvement with Martin Johnson and the senior England side, only for the RFU board to force a u-turn just 48 hours later.
But Thomas said Johnson has been unfazed by recent events at Twickenham.
"I have spoken to Martin and he is very clear in his response," he said.
"And that is 'I will do what is in the best interests of England, that is my sole objective'.
"He is incredibly focused on this World Cup."
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