The female was first spotted in North Berwick on Wednesday and is 31ft (9.5m) long, according to East Lothian Council.
A photo posted on the council’s Twitter account shows the beached whale lying on its side after the tide had receded.
The tweet said: “Unfortunately a dead minke whale has washed up at North Berwick.
“We are currently arranging for its removal but the public are advised to keep a suitable distance from the whale and not to allow dogs to go near the carcass.”
A council spokesman told the PA news agency: “Yesterday afternoon it had washed close to the harbour and then, with the movement of the tide, it went out to sea and came back in the morning time.”
The area around the whale has been cordoned off by police as the council makes arrangements for the carcass’s removal.
A statement posted to Facebook by conservation charity the Scottish Seabird Centre said: “Very sad sight this afternoon – a dead minke whale just off North Berwick harbour.
“This stranding has been reported to the Scottish Marine Animal Stranding Scheme (Smass), which collates and analyses all reports of stranded cetaceans and can carry out autopsies to understand more about the health and ecology of these fascinating marine animals.”
