Can we use higher strength SHA along with lower strength RSA key in SSL? I have noticed when the RSA public key is 2048 bit, SHA-256 bit used for signatures. When the key is 3072 bit, SHA-384 bit used.  When the key is 4096 bit, SHA-512 bit used.

I have tried the following commands.

openssl genrsa -out my.key 2048
openssl req -new -sha512 -key my.key -out my.csr


It worked without any errors.

My question is that, will there be any issues when the SHA-512 is used with 2048 bit?
In general, no, there is no security problem with using overkill that fits in your budget.

Maybe there are constraints in your application or protocol that would prevent you from using SHA-512 with 2048-bit RSA moduli, but we can't say without more details about your application or protocol.