The dollars, a four-figure sum, were found by staff at the museum in the High Street on Thursday..
Local officer, Constable Mark Richardson, Biggar Police Office, said: " We believe the money belongs to an Australian tourist and that he or she may have unknowingly dropped it when visiting the museum in the town on Wednesday."
On the police facebook page he added: "Despite several attempts, we have not been able to get a hold of the owner, who we believe may be en route to Edinburgh, so thought that an appeal on social media would reach far and wide and hopefully get to them.
"It's a considerable amount to lose and I am sure the owner will be, if not already, concerned when they realise it's gone, however, we hope to put a smile back on their face knowing it's been handed in."
The person who lost the dollars can contact Biggar Police Office on 101 quoting reference number 2071/03/08.
And chancers needn't try - the police have paperwork showing who the owner is.