Leicestershire club Market Bosworth RFC called for a special general meeting to vote on a motion that Thomas should resign from all executive and non-executive roles.
Thomas is currently the acting chief executive of the Rugby Football Union, England's representative on the International Rugby Board, the Six Nations committee and chairman of Rugby World Cup 2015.
For an SGM to be called, Market Bosworth's proposal requires the signatures of 100 of the RFU's member clubs.
But the RFU top brass have criticised the actions of some member clubs, jointly accusing them of acting against the "greater good" and betraying rugby's standards.
Murphy said: "It is obviously very disappointing that some council members have seen fit to propose an SGM with the sole purpose of removing Martyn Thomas from office.
"There can be no hiding away from the fact that those council members have put a personal matter above the greater good of the RFU and failed to recognise their collective responsibility that Martyn has won two votes of no confidence by a sizeable margin.
"At the AGM in July, Martyn stood down as chairman but rightly retained his other roles as he is highly valued by the international rugby community.
"The day to day operation of the RFU is not in a mess nor is it dysfunctional and we are all determined to look forward to the future opportunities of RWC 2015.
"Sadly it is a small section of the rugby community that are looking backwards."
The proposal to remove Thomas has been prompted by the conclusions of the Blackett Report into the removal of John Steele as chief executive.
Judge Jeff Blackett, the RFU's disciplinary officer, criticised Thomas' conduct and recommended he and the majority of the board stand down.
Wildash is overseeing an investigation into whether Thomas should be charged with misconduct following allegations contained within the Blackett Report.
Wildash has told the clubs that proposing an SGM before that process is complete is inappropriate.
"To call an SGM on the basis of a report containing allegations made anonymously and before the process has been completed to determine whether those allegations can be substantiated before anyone has been given the opportunity of defending themselves against any resulting charges, is surely not the way that rugby people should behave because it directly conflicts with our concept of fair play and justice," he said.
Thomas, who is in New Zealand at the World Cup along with Wildash and Murphy, insisted: "I remain committed in my role as chief executive until we have appointed my successor when I shall readily hand over and focus on my other roles as an International Rugby Board representative and Chairman of England Rugby 2015.
"I would also like to express my gratitude for all the calls of support I have received from staff at Twickenham and council colleagues."
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