Hill is currently recovering from injury, having not played since January, and watched the game with Sale's coaching team in the Recreation Ground's main grandstand.
As 6ft 7in Hill left his seat after the final whistle - Sale lost 31-23 - and walked down the stairs, he became involved with the man.
The Bath supporter, who did want to be named, told reporters: "I was sat there and he came down the stairs, grabbed me around the throat, broke my sunglasses and cut the top of my eye.
"He was like saying to me 'don't ever do that again'. To be honest, all I want is a new pair of sunglasses. I don't want to make a big thing. I was sat with my partner and all her family. This is not football."
In a statement, Sale said: "Sale Sharks are aware of an incident involving a player and Bath Rugby fan at the final whistle of today's semi-final.
"The club will work with Bath Rugby to complete an investigation into the incident and will provide no further comment at this time."
Asked about the incident in his post-match press conference, Sale rugby director Alex Sanderson said: "It is news to me. Jonny is not the aggressive type and I will answer those questions when I know.
"I can vouch for Jonny's character. He is not an overly aggressive, spiteful person."
