In the writ, Mowlem and Nishimatsu blame Atkins for the explosion, which was caused by a compressed air blow-out.
The writ says that Atkins failed to exercise proper skill or care, failed to consider the risks of using compressed air and did not carry out proper safety checks.
The explosion, which occurred on 23 February 1998 beside a primary school on the southern tip of the Isle of Dogs, left a crater on the surface and blew debris 150 m away from the blast.
The writ says John Mowlem Construction suffered losses of more than £1.2m, including the costs of civil engineering, additional design, supervision costs and delays in construction.
The writ added that Nishimatsu's losses amounted to more than £4.7m, including £208,769 for excavating the cofferdam and the tunnel and removing the debris.
The contractors are also suing Atkins for breach of contract and negligence following the blow-out. The companies believe that a single bolt pocket may have failed, which led to the tunnel unzipping progressively.
The High Court writ claimed Atkins failed to exercise proper skill or care
An HSE spokesperson said the preliminary case was heard at the City of London magistrates' court last month, but has now been referred to the Crown Court. The spokesperson said no date for the case has been set as yet.
The spokesperson added that Nishimatsu was being prosecuted under sections two and three of the Health and Safety at Work Act 1974. It is being accused of failing to oversee the safety of its employees and non-employees.
John Mowlem refused to comment on the writ.
An HSE spokesperson confirmed that a discussion document addressing safety concerns about crane climbing will finally be released and will deal with the lessons learned from the accident at Canary Wharf.