First Group said the offer “significantly undervalues” the business in a statement it noted had been made “without the consent of I Squared”.
The contingent right to the extra cash “does not provide shareholders with sufficient certainty”, added First Group.
I Squared is a US private equity fund based in Miami with assets of $13 billion and a focus on big infrastructure businesses such as energy and transport.
Founder Sadek Wahba is Egyptian.
First Group runs GWR and South Western trains. It used to run the yellow school buses in the US, but sold that arm and is now a UK business only.
Its shares were today at 136p, seemingly well below the value of the putative offer, suggesting City scepticism that I Squared has got much chance of success at the moment.
First Group said: “The board has unanimously rejected the proposal.”
I Squared did not immediately respond.