They say delays in school refurbishments by Jarvis Projects are disrupting education, affecting standards, and that the firm's work is shoddy. School standards minister David Miliband has ordered an urgent report.
Meanwhile, a former Burmese torture victim says a pound;34m Private Finance Initiative contract on North Tyneside should be withdrawn from Japanese construction company Kajima, which has links to the Asian military dictatorship.
Jarvis started work on Salendine Nook high school in Huddersfield in summer 2001. The school was due for completion in August but is still a building site, according to head Christine Spencer.
Mrs Spencer said schools in the three-year project had lost their initial excitement about the repair of their "dreadful" buildings. Overrunning work on science and IT labs, maths classrooms, and gyms had disrupted teaching, she said. Pupils missed days at the start of term. Heads also felt the workmanship was poor.
"You cannot have children without proper resources and still maintain high standards," she said. Her GCSE pass rates unexpectedly dropped 2 per cent this year.
Mrs Spencer said around 10 heads of Kirklees schools have sent evidence to the Government.
A statement from Jarvis and Kirklees Council, working in partnership on the project, said unforeseen problems were inevitable in such a complex project, but both parties were working to resolve them.
North Tyneside has named a consortium which includes Kajima Europe as preferred bidder to build and maintain four new schools in Whitley Bay and Wallsend.
Kajima has an office in Burma - now called Myanmar - whose military regime has been widely condemned for human rights abuses.
James Mawdsley, jailed in Myanmar for distributing pro-democracy leaflets, backs a Reject Kajima campaign.
It has 1,000 signatories, including, in a personal capacity, Peter Butler, president of the National Association of Schoolmasters Union of Women Teachers.
Mr Mawdsley, 29, said: "Kajima has supported the military regime in the past - their continued presence in the country shows they intend to do so in the future. North Tyneside should find a more honourable company to deal with."
Robert Longley, project director of Kajima Europe, said the office in Myanmar was an "accommodation address" and it had not done any work there for two years.
John Marsden, North Tyneside's chief executive, said that Kajima's bid offered schools high quality, best value and good delivery.
David Loveday, head of Burnside high school, said all four schools due to be rebuilt supported the Kajima bid.