I have a Chevy 307 with casting number 3970020 and everything I've read says this particular block can be bored out to 4" effectively making it a 327. I plan to have my machine shop test it and make sure this is a safe bore. If I do bore out the engine will I need different heads for it or will the 307 heads work fine?
tl dr; Yes. You can use the 307 heads on the build.
Now, with that out of the way, let's talk about what you'll get with these 307 heads. Every 307 head built came with 1.72" intake valves. These are relatively tiny. In fact, from the factory, these are the smallest intake valves offered (that I'm aware of). 350's came mostly with 1.94" intake valves.
The 307 was built small and maintains small numbers. You'll never get anything worth a bean out of these heads. If you want to build a 327, it would be much wiser to buy a 350, which will most likely have the larger 1.94" valves in the heads, and throw in a destroked crank. You'll get a far better building block than the 307 will ever be. Complete 350 engines are relatively cheap and still very plentiful.
If you are just wanting to go cheap, a good answer would be to get a pair of 350 Vortec truck heads. These are very common and really inexpensive for what you're getting. They have a great combustion chamber and flow well for what they are.
I don't know what your performance goals are for your 327, but with the stock 307 heads, don't expect miracles. The 307 was a low horsepower motor. Don't expect your 327 to be any different using these heads.