Police divers discovered the body near to the building yard at Horbury Bridge at around 12.50pm on Tuesday.
Formal identification has yet to take place but Mr Vasey’s family has been informed.
Detective Chief Inspector Fiona Gaffney, of Kirklees CID, said: “Following an extensive search for over the past four weeks, we are sad to inform the public that a body has been found in the search for Colin Vasey.
“While formal identification has yet to take place, the family of Mr Vasey are aware and they are being supported by officers.
“This wasn’t the outcome we were wishing for, but we hope Mr Vasey’s family will now have the closure they need and can begin to grieve for Colin. They have requested that their privacy is respected at this time.
“I would like to thank all those who have helped in the search for Mr Vasey – the outpouring of support and the response from the public has been nothing short of immense.
“We have had hundreds of people helping in whatever way they can, from providing and searching for CCTV, to on the ground searches. Their support is greatly appreciated by Mr Vasey’s family, the police and the wider community.”
Mr Vasey, 81, was first reported missing from Dewsbury on November 17.
Officers have been conducting extensive searches across the area since then, using drones, helicopters and specialist search officers.
There was also a huge response from the public to the appeals put out by police.
Det Ch Insp Gaffney said during the search: “The public response to this investigation has been enormous and really underlines his popularity locally, and the sympathy his disappearance has generated.”