The museum in Fort William’s Cameron Square is popular with fans of the Outlander books and TV series, who come to see its large collection of objects associated with the Jacobite cause.
Diana Gabaldon, the author of the books, toured the building last month while in Scotland to receive an honorary doctorate.
In 2012 the museum attracted about 9,000 visitors, but since then it has become free to enter and last year 59,000 people came to view its displays.
Nasmyth, who died in 1758, was a landscape and portrait painter and a friend of poet Robert Burns.
Travellers on a Country Path, Possibly Ayr Beyond depicts a scene in Burns’ home area of Ayrshire.
Lyon and Turnbull’s senior fine art specialist Alice Strang said: “What an amazing Girl Scout Amelia is, to have established not only a charity for her final scout project, but one that will support Highland heritage.
“Travellers on a Country Path, Possibly Ayr Beyond is a gem of a painting by one of Scotland’s most important artists and its sale will help the West Highland Museum’s exciting redevelopment plans.”