Cisco Systems is suing Apple Computer for trademark infringement in a US federal court, for using the iPhone name.
Apple launched its new handheld device under the iPhone name on Tuesday, at the Macworld event in San Francisco.
Following the launch Cisco said it hoped to resolve the matter by Tuesday evening after negotiations.
Apple responded by saying the lawsuit was "silly" and that Cisco's trademark registration was "tenuous at best".
"We think Cisco's trademark lawsuit is silly," Apple spokesman Alan Hely said. "There are already several companies using the name iPhone for Voice Over Internet Protocol (VOIP) products."
"We are the first company to ever use the iPhone name for a cell phone, and if Cisco wants to challenge us on it we are very confident we will prevail."
Cisco, which has owned the trademark since 2000, said it thought Apple would agree to a final document and public statement regarding the trademark.
"Cisco entered into negotiations with Apple in good faith after Apple repeatedly asked permission to use Cisco's iPhone name," said Mark Chandler, Cisco senior vice-president and general counsel, in a statement.
The firm has owned the name since it bought Infogear Technology in 2000, which registered the name.
Infogear had previously used the name for several years, said Cisco.
Cisco, which supplies networks, said it would seek to stop Apple from "infringing upon and deliberately copying and using" the trademark.
Jon Noh, a spokesperson for Cisco told the BBC: "This issue is not about money, and it's not about the phone itself; it is about Cisco's obligation to protect its trademark in the face of Apple using it without our permission."
Cisco's Linksys arm has employed the trademark since early 2006, and in December it launched its own phone, capable of connecting to the Internet, called the iPhone.
News of Apple's launch of its touch screen has been long awaited and Steve Jobs, the firm's head, said the phone would revolutionise the market.
Shares in Apple were 1.3% lower in after-hours trade, having risen 8% after the phone was launched a day earlier.