He was last seen leaving his home address in the Bessacarr area of the town shortly after 4pm on Saturday, December 4.
A statement from the force said: "We are sorry to share that a body has been found in the search for John from Doncaster. John, 84, was reported missing to us yesterday. Although formal identification has not yet taken place, we do believe the body is that of John and his family has been informed."
Speaking to its followers on Twitter, it added: "We know you have been really concerned and we have seen your comments this morning requesting an update, however we needed to ensure John's family had been told the news, and that they had had time to talk to extended family and friends before we published anything publicly.
"Thank you for all your help yesterday and overnight - we always appreciate your support."