You'll have a hard time finding an exact answer, as that's unlikely to be public information, but here are some general trends:
In the case of the high pressure turbine, the blades are one of the key company jewels. These are one of the highest technology pieces of the engine, and contributes a large part of the competitive advantage of one engine over another. Thus, this technology tends to be kept as a trade secret, and the parts are generally manufactured by the engine OEM themselves, and not outsourced.
For example, let's just say that tomorrow P&W discovered some new material that would increase EGT margin by 50 degrees. That would give them a huge advantage over GE. So they would want to keep it as closely guarded as possible. If GE ever found out how to make that material, the competitive advantage is gone. So they want to keep all of the information as closely guarded as possible. What's the best way to do that? You make everything yourself. If you outsource the manufacturing, then your material is out there in a place where you don't have control over it. One or two blade might "walk away", and suddenly show up somewhere else. You don't want that to happen. So you keep it all in house. In fact, you probably keep the information on a "need to know" basis. I.e. the engineers who design the compressor or the fan aren't told the details of this new material. They know it's there, and maybe some general information, but the details of how to make it aren't even widely known within the company.
Now that's for new engines. Ones that are actively being sold, where there is a competition, and customers are willing to pay a premium for performance. Now consider on the other hand a very old engine. Something like a CFM56-2, or a JT8D. That's ancient technology, so there's not really any competition on performance. But there are still a lot them flying around, and they do need spare parts from time to time. The key driver for the spares market is going to be cost. No one will pay a premium for performance here. So you might see some of the manufacturing outsourced in this case. You'll even see 3rd parties selling aftermarket parts for these engines (look up "PMA").