Is it possible to add sockets to items?

The Transmuter will take one unsocketed item (even magic or enchanted) + 2 gems of equal or greater level (to the unsocketed item) to consume the gems and place a socket on the item.

As Rilgon Arcsinh pointed out, there is a small chance for an enchanter to add a socket to an item when asked to enchant it. Of course, this will use up one of the limited number of enchants for the item. There is also a named enchanter, Jurik the Socketer, who is guaranteed to add sockets when asked to enchant an item, up to a maximum of two sockets.

It is very possible. One of the possible outcomes from visiting an enchanter is that you have a small chance of adding an additional socket, up to a maximum of four.

The cost will increase for each attempt, so make sure you have a fair bit of money, as you're just as likely to either add an additional enchantment, or get nothing at all.

There's an NPC that gives sockets. He's really rare. I found him in elemental oasis floor 1.