He held up the women, aged 31 and 37, as they used an ATM in Sauchiehall Street.
The man had a bladed weapon and demanded they hand over their cash.
But a passer-by intervened and the would-be mugger fled empty-handed.
The incident happened just after 2am.
Chief Inspector Alan Porte said: "He was a white male, of thin build and in his late 30s or early 40s. He was wearing a dark hat, gloves and a white scarf - with a tribal print on it - that covered his face.
"Given it is the middle of summer, his clothing would have made him very distinctive."