The incident happened at around 4.20pm on Monday, March 22 when the clothes of a man working at Crichness Law in the Lammermuir Hills caught fire.
It is understood that the man was a part-time casual farm worker who was burning heather.
Emergency services mounted a rescue operation and the man was taken by air ambulance to Glasgow Royal Infirmary.
The 54-year-old died there on Tuesday.
One other casualty was supported by the Scottish Ambulance Service at the scene. Their condition is not yet known.
Enquiries are ongoing to establish the exact circumstances surrounding the incident.
A police spokesperson said: “Around 4.20 pm on Monday, 23 March. 2021 a 54-year-old man was working in the Lammermuir Hills, Scottish Borders when his clothing caught fire.
“Emergency services attended and he was taken by air ambulance to Glasgow Royal Infirmary where he died earlier today, Tuesday, 23 March, 2021.
“His family are aware and a report will be submitted to the Procurator Fiscal.
“Enquiries are ongoing to establish the exact circumstances surrounding the incident and HSE has been informed.”
A Scottish Fire and Rescue Service spokesperson said: “We were called at 4.21pm on Monday, March 22 to assist our emergency service partners at Crichness Law, Cranshaws, Lammermuir, Scottish Borders.
“Operations Control mobilised three appliances and crews assisted a person in difficulty.
“One other casualty was supported by the Scottish Ambulance Service at the scene.”