Praise be to Allah.
It is not makrooh to purify oneself with Zamzam water according to our opinion. This is the opinion of all the scholars, apart from Ahmad according to one report narrated from him. Our evidence (the evidence for this) is that no saheeh report has been narrated to indicate that it is prohibited, and it was reported that the Prophet (peace and blessings of Allah be upon him) said: "Water is purifying and nothing makes it impure."
As for the reports that al-'Abbaas said it was forbidden to do ghusl with the water of Zamzam, this is not a saheeh report.