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I haven't directly seen the procedure, but I took a class focused at how it is done.
When a company needs a software project, they make public something like a manifesto, and software companies make offers.
Offers should include:
Dates when you will hand in advances
Additional equipment you or the company will need to buy
Sometimes the customers are themselves, that is, another branch within the company, in which case a manifesto is not needed.
Other software companies are focused on outsorcing, they take in other companies' ongoing software projects and add to them.
This is what I remember from the course, all this information might not be completely correct.
contacts and online advertisement...
In real life, the process works like this :
1. The software company will have a procurement team. They'll solicit projects from their current contacts or they'll respond to an advertisement from another company.
2. The software company will do a feasibility analysis basically judging if taking the project makes financial and strategic sense.
3. Once they think they can do a good job at it, they perform a full scale analysis outlining their plan ,the resources they'll need and the price which they'll charge the customer.
Most of the times , the software company will rely on its past projects success to highlight its strength and negotiation skills to grab the project.