It said the step was "inflammatory and inappropriate".
Railways Minister Ghulam Ahmad Bilour made the offer after a wave of protests against the film across Pakistan and the wider Muslim world.
Dozens have died in clashes between police and protesters in Pakistan and countries such as Tunisia and Sudan.
Sunday saw a fresh round of protests over the film in Pakistan, Nigeria, Greece and Turkey.
Most passed off peacefully, but in the Greek capital Athens, riot police and demonstrators clashed. Six people were arrested.
A state department official told the BBC: "The president and secretary of state have both said the video at the core of this is offensive, disgusting, and reprehensible - but that is no justification for violence, and it is important for responsible leaders to stand up and speak out against violence.
"Therefore we find Mr Bilour's announcement is inflammatory and inappropriate.  We note that the prime minister's office has dissociated itself from his comments."
The exact origins of Innocence of Muslims, the low-budget film that has prompted the unrest, are unclear.
The alleged producer of the trailer of the film, Nakoula Basseley Nakoula, is in hiding.
Anti-US sentiment grew after a trailer for the film dubbed into Arabic was released on YouTube earlier this month.
US citizens have been urged not to travel to Pakistan, and the US embassy has paid for adverts on Pakistani TV showing President Barack Obama and Secretary of State Hillary Clinton condemning the film.
Although US targets have borne the brunt of protests against the film, anti-Western sentiment has been stoked further by caricatures of the Prophet Muhammad published this week in the satirical French magazine, Charlie Hebdo.
France shut embassies and other missions in about 20 countries across the Muslim world on Friday.
An 18-year-old man who threatened the magazine's editors on Facebook has been charged with terrorism-related activity  following his arrest in the southern French city of Toulon, a  judicial source told AFP news agency on Sunday.

4 June 2014 Last updated at 21:59 BST
The government is so concerned about the lack of cover that it is introducing a new scheme that will force insurers to offer policies at affordable rates.
But that doesn't come in until next year.
Clinton Rogers reports from the village of Moorland.
The US state department has condemned a Pakistani minister's offer of $100,000 (Â£61,600) for the death of the American maker of an anti-Islam film.
[NEXT_CONCEPT]
Volunteers are rebuilding several uninsured homes on the Somerset Levels after floods that cost the West Country millions of pounds.