What's your line manager like? And why are you leaving your current job?
I have had a few employees ask me if they can put me down as a referee when they're looking for other jobs. I don't judge them for looking at all, because I don't really expect them to stay on in our organisation indefinitely anyway - we can't always offer a very defined progression route, so for someone who wants to develop their career, leaving is sometimes the only option.
When people tell me that they're applying for other jobs, I'll sometimes offer to proofread applications or do mock interviews. Assuming that they are good members of staff, it will certainly be a loss to us when they go, but my view is that helping people to achieve their personal career goals is part of what makes us a good employer and part of what attracts new talent to the organisation. People will always come and go, the important thing is that they contribute well while they're with us, and hopefully they learn a lot in the process that they can take along to their next role.
I'd hate to think someone felt that it would be awkward to share their plans with me. I'd much rather know, and I can then start planning for what to do if they leave. If they're not successful, it's sometimes possible to expand or develop their existing role to give them the experience that they might be lacking. Again, I don't judge - the fact that they haven't been hired for another role says nothing about their ability to do the job they're currently employed to do.
I guess what I'm trying to say is that you shouldn't necessarily assume it will be awkward telling your boss, as it might not be, and you never know, they might actually be able to help you.
It really does depend on your boss though. If you don't want to tell your current employer before getting an offer, I would avoid putting your boss's name down as a referee if you possibly can. I have been contacted in error on several occasions when an employee has said that they don't want referees to be contacted until after an offer has been made, and then the recruiting company has written to withdraw its request for a reference because they didn't have the employee's consent to contact me. Now that is awkward!