Former chairman Tom Ilube, part of the RFU's renumeration committee who approved Sweeney's recent bonus, stepped down in December.
His departure and the launch of an independent review of the scheme that boosted the pay of Sweeney and five further executives has not quelled discontent though.
If Sweeney's critics can provide the paperwork to show the necessary support, a special general meeting would be called in the midst of the men's Six Nations.
Several club officials explained their decision to support the no-confidence motion.
"The recent decision of the RFU to award bonuses to senior staff is beyond belief," said David Morton, secretary of sixth-tier Carlisle.
"The RFU keep trumpeting community rugby as the lifeblood of the game, while at the same time seemingly treating them like paupers."
"Every club I know wants to see change at the top, new leadership and a new approach to taking English rugby forward at all levels," added Paddy McAlpine, chairman of Chichester, who are also in tier six.
However, the nine professional referees employed by the RFU distanced themselves from the motion, saying they had not been consulted by the referees' union behind it and added they "have not endorsed any statements or meetings".