You are right but I just want to know if the following sentences are grammatically correct. Thank you.
He is reported to be sending her messages.
He is reported to have sent her messages.
He is reported to have been sending her messages.
He is reported to send her messages.
I think future forms are not possible, as the infinitive has to be used.
Your first three sentences are correct. The fourth one is not.
I think you're right that future forms cannot be used in this construction.