Among those on standby, are Barking, Havering and Redbridge University Hospitals Trust, Calderdale and Huddersfield Foundation Trust, Northern Lincolnshire and Goole FT, Leeds Teaching Hospitals Trust, and East and North Hertfordshire Trust, the HSJ reports.
An Environment Agency spokeswoman said: "The Environment Agency has found Health Environmental Services to be in breach of its environmental permits at five sites which deal with clinical waste.
"We are taking enforcement action against the operator, which includes clearance of the excess waste, and have launched a criminal investigation.
"We are supporting the Government and the NHS to ensure there is no disruption to public services and for alternative plans to be put in place for hospitals affected to dispose of their waste safely."
The leaked documents showed HES' Normanton site reached excess waste levels of 350 tonnes in September - five times its limit of 70 tonnes.
In an attempt to deal with the situation, it is said body parts being kept in fridges and the company is attempting to export 750 tonnes of pharmaceutical waste to Holland.
A Government spokesman said: "We are monitoring the situation closely and have made sure that public services - including NHS Trusts - have contingency plans in place. There is absolutely no risk to the health of patients or the wider public.
"Our priority is to prevent disruption to the NHS and other vital public services and work is under way to ensure organisations can continue to dispose of their waste safely and efficiently."
A spokesman for Healthcare Environmental Services said: "Healthcare Environmental has highlighted the reduction in the UK's high-temperature incineration capacity for the last few years.
"This is down to the ageing infrastructure, prolonged breakdowns and the reliance on zero waste to landfill policies, taking up the limited high-temperature incineration capacity in the market.
"Over the last year, this reduced incineration capacity has been evident across all of the industry and has affected all companies."
It added that it had "consistently highlighted" the issue to environmental regulators, and there has been no disruption to services to customers.