Answer
Wa `alaykum As-Salamu waRahmatullahi wa Barakatuh.
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:
Sending blessings on the Prophet (peace be upon him) while hearing his name mentioned in the recitations of the imam or while we are reciting those verses, is not a prescribed practice.
Answering your question, Sheikh Ahmad Kutty, a senior lecturer and an Islamic Scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
We are encouraged to send benedictions on the Prophet (peace be upon him) as often as we can; we ought to do it in salah, especially in the final sitting of every Prayer.
However, saying salawat on the Prophet (peace be upon him) while hearing his name mentioned in the recitations of the imam or while we are reciting those verses, is not a prescribed practice. If it had been prescribed, it would have been reported by the Prophet’s companions. We find no evidence for it in the sources.
Therefore, the scholars are divided on this issue; some permit or even recommend it as long as it is done silently and not repeatedly, while others say one should abstain from it.
No one, however, would disagree with the fact that saying salawat on the Prophet (peace be upon him) is one of the most meritorious acts when practiced right at the right time and place.
Allah Almighty knows best.