Answer
In the Name of Allah, the All-Merciful, the Mercy-Giving
All praise is due to Allah. Peace and blessings be upon His Messenger Muhammad.
In this fatwa:
You can help your uncle perform Umrah and you will be greatly rewarded for this, but you can not pay for Umrah with your zakah.
In his response to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
You are not allowed to give zakah to your poor uncle to perform Umrah as it is not obligatory on him.
Zakah should be given only to one of the eligible categories as stated in the following verse:
{Compulsory arms are (meant) only for the poor and the needy, and those who are charged with collecting them, and those whose hearts are to be won over, and for the freeing of human beings from bondage, and (for) those who are overburdened with debts, and (for those who strive ) in Allah’s cause (fi sabil Allah), and (for) the wayfarer: (this is) an ordinance from Allah–and Allah is all-knowing and, all-wise.} (At-Tawbah 9:60)
However, you can help your uncle perfrom Umrah and you will be greately rewarded for this, but you can not pay for umrah with your zakah.
Almighty Allah knows best.
Source: www.askthescholar.com