The academy, based in south London, requires expressions of interest to be submitted by 14 November.
The brief includes PR and comms support as well as stakeholder engagement with politicians and the education sector.
The incumbent agency is PLMR, which is working on an ad-hoc, not-for-profit basis, after its existing contract with the academy ended in August. It had been working on a for-profit basis for the academy since May 2009.
PLMR was told earlier this year that it cannot re-tender for the contract on a for-profit basis, although other agencies can.
The agency, which won two Cannes Gold PR Lions in 2011 for its work lists public affairs, crisis communications and media relations among its functions but its relationship with the academy is controversial.
PLMR’s managing director, Kevin Craig, sits on the board of the academy’s governing body, a move which was highlighted by the Education Funding Agency watchdog as a concern.
The EFA also said that a contract for a company owned by Sir Greg Martin, the academy’s executive head, to manage the academy’s leisure facilities must end because of the "risk of a conflict of interest".
In a letter to the academy’s chair of governors earlier this year about Craig’s appointment and the contract with Martin’s company, the EFA said the arrangements were "not sufficiently transparent or fully compliant with the latest Academies Financial Handbook."
The EFA told the academy it must carry out a full open tender for the PR contract when the current one expired in August.
It said PLMR could then end its involvement with the governing body, not bid, or bid on a not-for-profit basis.
Elin Twigge, deputy managing director at PLMR said: "We have worked on this for a long time. It’s an amazing school and we will consider tendering on a not-for-profit basis."
Twigge said this would involve providing its services "at cost" to the academy.
The academy has used procurement websites to invite tenders for the new contract but with limited response, including one that is offering its service on a ‘for-profit’ basis.
But Julian Geary, the academy’s finance director, told PRWeek that PLMR’s likely not-for-profit bid "did not lock people out" of the tendering process because it would be seeking "best value" rather than the best price.
Geary dismissed suggestions that the tendering process was a foregone conclusion but he admitted that PLMR’s existing relationship with the academy would have a bearing on its decision.
He said: "If we’re not dissatisfied with PLMR, and we’re not, its bid would be of great interest, to say the least."