The move is being led by quantity surveyor Jeremy Hackett. He is calling for an extraordinary general meeting and another vote on the fee increase.
Hackett said: "Soundings have been taken to organise an EGM in the autumn, for which we would need about 860 member votes. Many of the members I have spoken with are interested in tabling a motion to repeal last week's vote."
RICS members voted for the fee increase last Wednesday by a majority of 337 in a poll in which more than 17,000 voted. It will mean a rise in annual membership subscriptions from £190 to £250. Fellows will pay an extra £104 in addition to a subscription of £250.
Responding to Hackett's call for an EGM, RICS president Peter Fall said: "If he feels he needs to call another meeting and has the required support, he is entitled to do so. But the vote has been taken and it seems pointless to do the whole thing again."
I am increasingly dubious about the direction the RICS is taking
Jeremy Hackett, RICS member
The RICS says the increases are necessary to support the institution's Agenda for Change programme, established in 2000. The aim of this initiative is to raise the public profile of surveyors and to develop the RICS globally. It was initially funded from RICS reserves but these have been depleted by falls in the stock market.
Last week's meeting revealed widespread dissatisfaction among RICS members, who feel the body is becoming increasingly inaccessible. Hackett told fellow members during the debate: "I have been a faithful member of the RICS since 1966 and have supported the institution through thick and thin, but I am increasingly dubious about the direction it is taking.
"There is huge anger among the old QS division that the RICS is moving away from its core membership to focus on an international agenda and bigger businesses. We are no longer getting value for money."
Edward Reeve, a chartered surveyor in London, said he felt the RICS had failed to deliver on its promise to raise the profession's profile. He said: "The institution has done absolutely nothing to assist my career and, despite the Agenda for Change programme, chartered surveyors are still considered something of a joke."