Reference: Fataawa al-Hajj wal-‘Umrah – Page 18
Question: Regarding making the intention upon undertaking the rites [of hajj or ‘umrah], is it required to audibly pronounce it when uttering the talbiyah?
Response: It is from the talbiyah to utter:
«Here I am [in response to Your call Allaah] making ‘umrah»
…if one is intending [to perform the] ‘umrah, and:
«Here I am [in response to Your call Allaah] making hajj»
…if one is intending [to perform the] hajj.
As for [making] the intention, then it is not permissible to audibly pronounce it; So, [for example] one must not say:
«اللهم إني أريد العمرة»
«Oh Allaah! I intend [to perform] the ‘umrah», or:
…«…the hajj»; [saying] this has not been [authentically] reported from the Prophet (sal-Allaahu ‘alayhi wa sallam).