A cabinet meeting heard the amount equates to £220 per taxpayer and the estimated bill was dubbed "not affordable".
It comes after the Grenfell Tower fire in London and as the council works to remove cladding from two buildings.
Only 75 out of 2,523 high rise and sheltered dwellings in the city currently have sprinklers.
A report to the meeting said the cost of sprinklers could not be covered in the council's approved Capital Programme.
Councillors were warned the upgrade could take several years due to "capacity constraints in the industry" following the London fire.
They agreed to start the procurement process while lobbying the government for funding.
Council leader Donna Jones said the authority needed a fully-costed programme of work in place, in anticipation of a government invitation to bid for funds.
She said sprinklers might not be needed in all council residential buildings.
Dave Curry, chief officer of Hampshire Fire and Rescue Service, urged authorities to fit sprinklers, saying it was like "having a firefighter in every room in your home".
Following his statement Southampton City Council revealed it would be "retro-fitting" sprinklers in some of the authority's high-rise blocks in the next month.
Cladding on two of Portsmouth's 13 high rise blocks - Horatia House and Leamington House - is being removed after failing fire safety checks on 22 June.
The council said all of its tower blocks complied with fire safety legislation.
Portsmouth City Council is facing a potential £12.2m bill to fit sprinklers in tower blocks and sheltered homes.