It claims that iPhones using chips by rivals, such as Intel, infringe six of its patents.
Qualcomm said it had asked the US International Trade Commission (ITC) to investigate and impose an import block.
It is the latest move in a series of disputes and lawsuits between Apple and Qualcomm.
In January, Apple filed two lawsuits against Qualcomm, claiming it had abused its dominant market position.
It also alleged the chip-maker had broken an agreement between the two companies, by denying Apple access to chip technologies it was entitled to use under the terms of a licensing deal.
The latest disagreement concerns six patents relating to energy-saving features on the iPhone, which Qualcomm says Apple has used without permission.
According to the news agency Reuters, Qualcomm has also filed a lawsuit seeking financial compensation from Apple.
"Apple continues to use Qualcomm's technology while refusing to pay for it. These lawsuits seek to stop Apple's infringement of six of our patented technologies," the company said in a statement.
Apple referred the BBC to a previous statement on Qualcomm, in which it said the company's business practices "harmed the entire industry".
"They supply us with a single connectivity component, but for years have been demanding a percentage of the total cost of our products - effectively taxing Apple's innovation", it said.
"We believe deeply in the value of intellectual property but we shouldn't have to pay them for technology breakthroughs they have nothing to do with."
Qualcomm said it hoped the ITC would begin an investigation in August.
Qualcomm, the world's biggest producer of mobile phone chips, has appealed for the sale of some iPhones in the US to be blocked.