I am looking for a software that is a robot and will browse the web
automatically using a few seeds that I will give.

The software will look for certain words or patterns in the SOURCE
code of html documents of the sites it visits.
It will then create a list of the sites that match the criteria.

For example, I may want to find sites that have the following pattern
in their html source code: <a href=://www.google.com> to find a
list of sites that link to google.

This software may run on Linux or Windows, I don?t care.
Software should be FREE or have a FREE trial that doesn't limit its
functions so I can test it out.

Not really an answer but it could also help you find the necessary files:
- Offline Explorer (http://www.metaproducts.com/ - currently having
some technical problems... - free trial) will allow you to download
hundreds of pages
=> afterwards use Windows' search function to find the key in the downloaded files.

It would also interest me if such a tool as you are looking for really exists...