The coastguard agency said a helicopter, fixed-wing aircraft, lifeboats and nearby vessels with firefighting capability had all been called to the incident to help.
Thirty-two casualties were brought ashore with ambulances waiting to take them to hospital in Grimsby, the chief executive of the Port of Grimsby East said via email.
It was not clear how severe their condition was.
Local MP Graham Stuart said he had been told by the transport secretary that 37 crew members were aboard the two ships, and one was hospitalised.
"The other 36 mariners across both crews are safe and accounted for," he said.
The vessels involved are the US-flagged 49,729 deadweight tonnage (dwt) tanker Stena Immaculate and the Portuguese-flagged 9322 dwt container ship Solong.
The tanker is one of 10 in a US government program designed to supply the armed forces with fuel when required.
Television images from the BBC showed at least one vessel ablaze with clouds of black smoke billowing into a grey sky.
The Stena Immaculate and Solong were alongside each other off the coast, according to the last AIS ship tracking position update at 10.34am, LSEG shipping data showed.
The Stena Immaculate was at anchor near the port of Grimsby on Monday morning after sailing from Greece, according to ship-tracking site VesselFinder.
The cargo vessel Solong was sailing from Grangemouth in Scotland to Rotterdam in the Netherlands.
Stena said its tanker was operated by US logistics group Crowley.
Crowley wrote on X that the tanker, carrying a cargo of Jet-A1 fuel, was struck by the Solong while anchored near Hull off the North Sea coast.
"The Stena Immaculate sustained a ruptured cargo tank," Crowley said.
"A fire occurred as a result of the allision and fuel was reported released," the company said, referring to when one vessel is stationary.
The crew of the Stena Immaculate abandoned the vessel following multiple explosions onboard and all mariners were safe and accounted for, Crowley said.
"The Stena vessel is a products tanker. Pollution risk less than if it were a crude carrier," one insurance specialist said.
"A lot depends really on cargo carried, how many tanks were breached and how bad the fire is."
It was too early to assess the extent of any environmental damage, a spokesperson with environmental group Greenpeace said.
The incident occurred in a busy stretch of waterway, with traffic running from the ports along the United Kingdom's northeast coast to the Netherlands and Germany, shipping industry sources said.
"There were reports that a number of people had abandoned the vessels following a collision and there were fires on both ships," the Royal National Lifeboat Institution (RNLI), a lifeboat service working on the emergency response, said.
Maritime analytics website Marine Traffic said the 183-metre-long Stena Immaculate was anchored off Immingham, northeast England, when it was struck by the 140-metre-long Solong, which was en route to Rotterdam.
Ship insurer Skuld of Norway would only confirm that the Solong was covered with it for protection and indemnity, a segment of insurance that covers environmental damage and crew injuries or fatalities.
The United Nations shipping agency, the International Maritime Organisation, said it was aware of the situation.
with PA and AP
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