"The collision between USS Hartford (SSN 768) and USS New Orleans (LPD 18) occurred at approximately 1:00 am local time (2030 GMT on Thursday)," the Bahrain-based Fifth Fleet said.
"Fifteen sailors aboard the Hartford were slightly injured and returned to duty. No personnel aboard New Orleans were injured," it added.
The New Orleans' fuel tank ruptured, causing the spill of diesel fuel.
Both ships are operating under their own power, and the overall damage to both vessels is being evaluated, the statement added.
The Strait of Hormuz, less than 100 kilometres (60 miles) at its widest point, separates Oman from Iran and is the gateway into the oil-rich Gulf.
An estimated 40 percent of the world's crude oil passes through the strait on the way to market.
Both the submarine and the ship are on regularly scheduled deployments to the US Navy Central Command area of responsibility, the navy said.
Ships from the Fifth Fleet, based in Bahrain, patrol a 7.5 million square mile (19.4 million square kilometre) area of eastern Africa, the Middle East and Southwest Asia.