Qualcomm filed the suits in a Beijing intellectual property court, claiming patent infringement and is seeking injunctive relief, Bloomberg said, citing a Qualcomm spokeswoman.
Apple's shares were up marginally, while Qualcomm shares were little changed in afternoon trading.
Apple is waging a global legal battle on Qualcomm's long-held practice of charging a percentage of the total price of iPhones and other Apple devices as a licensing fee for its patents.
"Apple believes deeply in the value of innovation, and we have always been willing to pay fair and reasonable rates for patents we use. In our many years of ongoing negotiations with Qualcomm, these patents have never been discussed and in fact were only granted in the last few months," the company said in a statement.
"Regulators around the world have found Qualcomm guilty of abusing their position for years," the statement said. "This claim is meritless and, like their other courtroom maneuvers, we believe this latest legal effort will fail."
Qualcomm did not immediately respond to requests for comment.