How to build a WP site offline

I have just started using WordPress to build sites for my clients. Many of my clients have an existing website but they want me to create a new one for them. They then want my new site to be live under their existing domain name.

**This is not a question about transfers, hosts, nameservers etc – I know how to do that and have done it successfully several times.**

This is a question about whether or not there is a way to create the new sites for each of my clients BEFORE I transfer their domain, hosting, nameservers etc?

Basically, I need to be able to show them a preview of their new website, fix it and tweak it to their satisfaction…. the whole time leaving their existing, not so loved, website live for the public. But I really don’t want to have to purchase a pile of temporary domain names to be able to do this.

So is there any way to build a WordPress site BEFORE you have an actual domain to work with?

I’ve read several posts that have attempted to ask this type of question, but never seemed to express themselves clearly enough to get the answer they were looking for…. hoping that my explanation of what I’m asking makes sense.

I’m wondering if that’s what the “local” install of WordPress might be for, but wasn’t sure.