David McLeod, 78, was reported missing after he left his home in Woodstock Avenue, Galashiels, at around 8pm on December 16 – saying he was going to buy a newspaper and did not return.
A search was launched for Mr McLeod, who police said had been diagnosed with dementia and may have become confused.
Police Scotland said on Sunday that a body has been found in the Cauldshields Loch area.
Formal identification is yet to take place, however Mr McLeod’s family has been informed.
There are not thought to be any suspicious circumstances.
A report will be sent to the Procurator Fiscal.