Praise be to Allah.
The basic principle is that it is not permissible to sell water so long as it is still in its spring or stream. But if someone collects it and puts it in a vessel, for example, then he may sell it and there is no difference of opinion among the scholars.
Ibn Qudaamah (may Allah have mercy on him) said in al-Mughni (4/215):
With regard to what he collects in his vessel of water, he takes possession of it thereby and he has the right to sell it, and there is no difference of opinion among the scholars. … Based on that, it became customary in some regions to sell water in vessels without anyone objecting to that, and no one has the right to drink from it or to do wudoo’ or to take any of it without the permission of its owner. The same applies if he comes to his own well or a public well and draws water in a bucket or by any other means; whatever he collects of water becomes his property and he may sell it, because it became his property when he acquired it in his vessel. Ahmad said: It is forbidden to sell water that is still in the well and water that is still flowing in the spring, but it is permissible to sell the well itself, or the spring, and the one who buys it has more right to its water.
Shaykh al-Fawzaan (may Allah preserve him) was asked: Is it permissible to sell water and when?
That is subject to further discussion. If a person has collected water in his vessel or in his pond, then he has taken possession of it and it is permissible for him to sell it, because he has collected it and it has come into his possession, and he has expended effort to acquire it, so it has become his property.
But if the water is still in the well or river or stream that runs through his property, then this is the subject of a difference of opinion among the scholars. The correct view is that it is not permissible for him to sell it; rather he is more entitled to benefit from it than anyone else, but he does not have the right to prevent others from benefiting from it in ways that do not harm him or his property, because the Prophet (blessings and peace of Allah be upon him) forbade selling surplus water. End quote.
No distinction is made in this regard between Zamzam water or any other water.
Shaykh Ibn Baaz (may Allah have mercy on him) said:
There is nothing wrong with selling Zamzam water or transporting it from Makkah. End quote.
Majmoo‘ Fataawa Ibn Baaz (16/138)
And Allah knows best.