The fire authority exchanged contracts with developer Hadston earlier this month.
Hadston – said to be backed by US-based private equity firm Westbrook Partners – plans to turn the site into housing as well as hosting a new secondary 'free school'.
The £53 million price tag is said to be less than the market value due to the cost of providing the school on part of the site.
Valuers GL Hearn have confirmed that the deal represents 'best consideration' under local government law.
According to be a report to be considered by LFEPA members on Friday, the sale price will rise from £53 million to £54 million if the London Fire Brigade Museum does not stay at the Southwark Bridge Road site.
A decision on the future location of the museum is expected in July.
Stephen Knight AM – a Lib Dem member of LFEPA – said: "I hope this deal does deliver the new school that Southwark pupils so desperately need, however the contract in no way guarantees this outcome.
"Given that the Mayor intervened in the sale of this site for the express purpose of delivering a school, it shows reckless naivety that he refused to allow the fire authority to make the deal conditional on a school being delivered.
"We were told that the developer, and its US-based private equity backer, wouldn't do a deal if it was conditional on delivering a new school, which must beg some serious questions about their commitment to the new school and certainly set alarm bells ringing amongst members of the fire authority.
"The Mayor's stance is simply to hope for the best that a school will be provided, despite the fact that the developer must have a big financial incentive to do the opposite."