The earliest recorded use of 'smell a rat' meaning 'be suspicious' was in a poem by the Poet Laureate, John Skelton (1460 - 1529) complaining of how the clergy could argue that black was white and make ordinary people believe anything.
On the other hand, he claimed they could be as vicious as a cat who - on the slightest hint of the smell of a rat - would catch and treat the creature mercilessly. The clergy, of course, at that time were perfectly happy to burn at the stake anyone who showed the slightest doubt about their claims.
So, the phrase has nothing to do with US smelling actual rats, but rather about a comparison between cats and powerful clergymen.