Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
There is nothing wrong in pronouncing the intention while offering sacrifice (udhiyah).
In responding to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
It is not necessary to mention the intention of udhiyah aloud when slathering the sacrificial animal so as the intention is in mind. As Imam Ibn al-Qayyim said, “the locus of the intention is the mind.”
Therefore, it is enough to say: bismillah Allahu akbar with the intention of sacrifice. However, we read in the tradition from Aishah that the Prophet (peace be upon him) sacrificed the sheep by saying: In the name of Allah. O Allah, accept this sacrifice from me on behalf of Muhammad, his family and behalf of the Ummah of Muhammad.”
Based on this, there is nothing wrong in pronouncing the intention while offering sacrifice.
Almighty Allah knows best.