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.
《水浒传》中那位赤手空拳的打虎英雄武松,生活中确有其人。他是《水浒传》作者的好友卞元亨。据《盐城县志》、《古盐卞氏族谱》记述:元末两淮盐运副使卞仕震的儿子卞元亨,家住盐城便仓,年轻时候臂力过人,那时便仓一带常有老虎出没,当地人没有敢单独出行的,有一次卞元亨一人出行,遇到老虎,赤手空拳将老虎打死。
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.
历史上有这么一个武松,他原是浪迹江湖的卖艺人,曾为杭州知府中的提辖。后来杭州知府是太师蔡京的儿子蔡鋆,虐政殃民,人称蔡鋆为“蔡虎”。武松对这个奸臣恨之入骨,决心为民除害。他身藏利刃,隐匿在蔡府之前,刺杀了蔡鋆,但被官兵捕获,惨死于狱中。
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.