All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
After investigation, we have not come across a Prophetic narration about the excellence of performing 'Umrah on Friday in Ramadan. However, there is a Prophetic narration which reads "An 'Umrah in Ramadan equals Hajj'' [Al-Bukhari and Muslim]. Another narration reads: "An 'Umrah in Ramadan equals Hajj with me" [Al-Bukhari], this applies to the entire month of Ramadan and not only on Fridays during this month.
Therefore, you may perform 'Umrah at any day in Ramadan and you will get the above reward, Allaah willing.
Finally, it should be noted that a person should not believe in the excellence of some acts of worship if they are performed at particular days if the Islamic legislation did not specify any excellence for such acts during those days.