The deadline for the transition is 30 September 2006, but the US department has the option to delay the hand-over.
Porn domain
Icann, a California-based not-for profit company, was given the task of coordinating and managing the domain name system in 1998.
This includes the allocation of internet protocol numbers, the unique number given to every online device, as well as the assignment of domain names and deciding whether root-level names such as .com, .net or .org, can be added to the internet.
Currently, any major changes to the domain name system need to be submitted for approval to the US Commerce Department, but a move to a fully private system will mean the US government will relinquish this control.
Some are worried the US government has too much of a hold over the internet.
In May 2006, Icann rejected a plan for a new .xxx domain name, for sexually explicit sites, a move that some condemned as politically motivated.
ICM Registry, the company who wished to register the .xxx domain, is suing two US government departments for access to documents it claims show how they lobbied for the new net address to be rejected.
The hearing on Wednesday, will hear evidence about whether Icann is ready to stand free of the government.
Steve Del Bianco, director of The NetChoice Coalition, a Washington policy group, told Reuters: "The incentive (for the US to privatise Icann) is to keep the internet on one domain name system to avoid multiple systems - much like the multiple phone systems we have around the world."