There appears to be two theories that I can find regarding Wu Song's origin.
The first theory (source, source) seems to suggest that Wu Song's deeds were based on the bare-handed tiger slaying of Water Margin author Shi Nai An's friend and cousin Bian Yuan Hen.
The hero Wu Song who killed a tiger with his bare hands in Water Margin existed in real life as Bian Yuan Hen, the friend of the author of Water Margin. According to the Yan City District Report and the Ancient Yan City Bian Genealogy Chart, in the waning years of the Yuan Dynasty, the son of Bian Shi Zheng, Bian Yuan Hen of Yan City Bianchang Region was extremely strong when young. There were tigers in the region which made nobody dare to travel alone. Once, Bian Yuan Hen travelled alone, and killed a tiger with his bare hands during his travels.
The second theory (source, source) seems to suggest (as per Semaphore's comment) that Wu Song was an assassin who killed a local official (nicknamed Cai "Tiger") and was then tortured to death in prison.
There was such a Wu Song in recorded history. He was originally a roaming bard, who once worked for the Hangzhou magistrate office. The Hangzhou magistrate was replaced with Cai Yun, the son of the Prime Minister Cai Jing. He was a tyrant who abused his people, and was given the nickname Cai "Tiger". Wu Song hated this evil official, and decided to assassinate him. He carried a blade and hid at the door of the Cai residence, successfully assassinating Cai Yun. However, he was captured by soldiers and died in prison.
A combination of both theories, either or neither could be correct. I was unable to find more definitive primary sources, but again Chinese academic sources on ancient literature is much more difficult to find online than English literature academic sources, even when searching in Chinese.