The founder of the seized Megaupload websites is seeking to have the hearing broadcast on YouTube.
The US - which wants New Zealand to hand over Dotcom and three former colleagues - has opposed the proposal.
The request was made by Dotcom's lawyer on the first day of the appeal hearing in Auckland, which is expected to last up to eight weeks.
The FBI took control of Megaupload.com and other domain names belonging to the business in January 2012. Federal prosecutors accused the files-sharing site of having cost movie studios, music labels and other copyright-holders more than $500m (Â£382m) in lost revenue.
Last December, a New Zealand court ruled that the German-born entrepreneur could be extradited to face charges of copyright infringement, racketeering and money laundering.
Dotcom's lawyer, Ron Mansfield, claims the site should not be held responsible for the actions of its users, and said that his client did not get a fair hearing last year.
Mr Mansfield said there were "unprecedented issues of public and international interest" raised by the case and added that coverage should not be limited to traditional media.
The defence lawyer suggested that there could be a 10-minute delay to an online stream to let the court prevent any sensitive details being broadcast.
The High Court judge, Justice Murray Gilbert, criticised the fact the request had not been made in advance but said he wanted to hear the views of local media outlets before making a decision.
In a related development, it has emerged that one of Megaupload's websites is now hosting advertisements for pornography.
The news site Torrentfreak reports that an unknown party appears to have "hijacked" the domain despite it still being registered to the FBI.
A judge in New Zealand is considering Kim Dotcom's request to have his extradition appeal livestreamed online.