The demand follows the death of 12-year-old Keane Wallis-Bennett who was killed after a "modesty wall" fell on her at the city's Liberton High School last week.
Almost 7000 pupils in the capital are being taught in schools that have or are close to showing major defects.
Education leaders said they would bring classrooms up to at least a "satisfactory" standard by 2019. But critics, including senior politicians and parent representatives, have said the timeframe is unacceptable and are urging council executives to speed up the repair works.
Ian Murray, MP for Edinburgh South, said: "The council is strapped for cash, as are most organisations.
"If it needs £30m then the only way it could possibly do this quicker - which is what we want - is to get more resources.
"The only way it can do that is to get them from the Scottish Government. This is a real opportunity for the council and the Scottish Government to work together to identify additional resources to ensure this five-year timescale is minimised."
A joint inquiry by Police Scotland and the Health and Safety Executive into Keane's death is about to enter its second week.
A wall at Craiglockhart Primary has now been demolished as part of a city-wide sweep of structures similar to the one that killed the pupil.
Councillors promised to do "everything in their power" to prevent a repeat of last Tuesday's tragedy.
Jason Rust, education spokesman for the Edinburgh Conservatives, said: "I think the main focus has to be the inquiry at Liberton.
"But if more checks are being carried out at other establishments in the city then that is welcome. I think the next steps will all depend on the outcome of the inquiry."
Melanie Main, education spokeswoman for the Edinburgh Green Party, said: "The improvement programme was always a compromise between what is needed - much more - and what was available, so it is right to look again at the timing of the works.
"Bringing forward borrowing a year or two should be achievable, so needs to be seriously considered."
Lindsay Law, parent representative on the city's education committee, said: "If these works are required the council needs to seek additional funding from central government.
"If the timescales are being affected by the fact it does not have enough money then it needs to look somewhere else for that money.
"This is too important to leave it for years and years while things deteriorate further."
Sue Bruce, the council's chief executive, said more than 200 council facilities had now been inspected, including all primary, secondary, special and nursery schools, plus sports, leisure and community centres.
Government ministers acknowledged public demand for robust action but said the police and Health and Safety Executive investigation should remain the immediate focus.
A Scottish Government spokesman added: "We are already putting significant investment into the £1.25 billion Scotland's Schools For The Future programme. This is undertaken by working in partnership with local authorities to tackle the worst condition schools.
"This is in addition to capital allocation for councils and is being used to directly fund new schools and the refurbishment of existing schools."