It is understood that the project is eight months behind schedule and £700,000 over budget. McAlpine and R&H are at loggerheads over who is to blame for the extended construction programme and increased cost.
Work on the high-tech biological sciences laboratories on Tennis Court Road started in July 2001 and was due to be completed in July 2003.
Sir Robert McAlpine is believed to have issued at least 150 record-of-event notices for the project, each citing specific problems that may have delayed the building work.
A non-completion order has been issued and a deadline was set for 20 February. However, this date has since been pushed back to May – a 10-month extension to the original completion date.
The laboratories will form part of the Wellcome Trust–Cancer Research UK Gurdon Institute, part of Cambridge University, and will be used for research into cancer and developmental biology, as well as for teaching.
The laboratory is being funded by the Gurdon Institute's two charitable sponsors and the university.
Scientists at the university are concerned that the institute might have to bear some or all of the extra cost, and that this might affect its ability to fund research.
Sir Robert McAlpine, R&H Partnership and the University of Cambridge all refused to comment.
n Cambridge University this week shelved proposals to build a multimillion-pound animal-testing centre. The centre, which was to be built on green-belt land at Girton, two miles outside the town, was intended to attract experts from around the world to carry out research into neurological conditions.
A university spokesperson said the decision to abandon the plans was taken because of the "financial risks" involved. These included the cost of ensuring the centre was protected from animal rights protesters.
The nearby Huntingdon Life Sciences laboratories have experienced a number of highly publicised protests against vivisection.