Among the options being considered by the society are the offer of free or cheap access to its members, possibly restricting the service to certain customers.
But it has not ruled out mounting a direct challenge to service providers, such as Pipex, Virgin or Demon, by marketing cheap access to the Internet to all potential clients.
Its move raises the prospect of a bitter price war, as increasing numbers of institutions move to challenge existing providers by offering access as a simple bolt-on service.
Nationwide's move comes as increasing numbers of UK financial services institutions have set up "home pages" on the Net, providing information about their services to surfers. These include share information, mortgages, bank charges and fund managers' performances.
Barclays bank has gone one stage further, by setting up an on-line "shopping mall" through which one can visit a number of stores and purchase a range of goods.
Last month, Royal Bank of Scotland announced that it will offer banking on the Internet, claiming that it had cracked the problem of providing adequate security for users.
Up to now, the traditional mechanism has been to use a service provider to display a site on the Net, which can then be visited by any number of surfers, who themselves access the Net via a service provider.
The standard charge for access to the Net ranges between pounds 8 and pounds 12 per month, with callers paying additional phone charges at a local rate.
A Nationwide spokesman said the plans followed its move to expand its PC banking operation, which doubled to 10,000 after a simple mail- out. No firm decision has yet been taken.