Prices will start at £98 for a refurbished PC, with subsidised net connections available for £9 a month.
The 12-month trial is part of the Race Online 2012 scheme, which aims to reach out to the 9.2 million adults in the UK who have never used the net.
Distributor Remploy hopes to sell 8,000 machines in the next 12 months.
"Motivation and inspiration are still two of the biggest barriers [to using the internet], but clearly perception of price is another big deal for people," Martha Lane Fox, the UK's digital champion, told the Financial Times. "A good price point is certainly part of what helps people get online."
Race Online 2012, which aims to "make the UK the first nation in the world where everyone can use the web", estimates that of the more than nine million adults in the UK who are currently not online, four million are socially and economically disadvantaged.
The cheap computers will run open-source software, such as Linux, and will include a flat-screen monitor, keyboard, mouse, warranty, dedicated telephone helpline and delivery.
The packages will be sold through 60 UK online centres which offer IT training and Remploy, an organisation that specialises in helping disabled and disadvantaged people find work and which runs the computer recycling scheme e-cycle.
Race Online 2012 has also negotiated cheap internet packages using a mobile dongle, costing £9 a month or £18 for three months, to help people access the web.
Its research suggests that going online can save people around £560 a year and that thousands of jobs are offered exclusively online.
But the cost of owning and running a computer and net connection is often seen as a barrier for many people.
As a result, there have been several previous government-sponsored initiatives that offered cheap PCs.
The £300m Home Access Scheme began to distribute free laptops to pupils from poor backgrounds in January 2010. It was scrapped by the coalition government eight months later.
Low-cost computers are to be offered as part of a government scheme to encourage millions of people in the UK to get online for the first time.