The relation between Izaya and Namie borns in the moment in which Namie asks to him to find Seiji (that has escaped with Mika) for her.

Izaya proves to know her well. He knows that she's a brilliant profesionist that reached her goals working hard, a strong person, and, than... a beautiful woman. These are Izaya's textual words, eheheh!
Obviously, he's informed because of Namie's position in Yagiri Pharmaceuticals (a company that acts experiments on humans), but the comment about her physical appearance is TOTALLY personal.

He's sarcastic and ironic with her, but she understands at once Izaya's personality, and she says to him to shut up everytimes he says anything she doesn't approve.

When Namie escapes with Certy's head, she has no idea of what she can make in order to doesn't finish in prison, and again, she asks help to Izaya, that proposes to her to works with him.
So they start to work together and to hide Certy's head.

The relation between them becomes deeper.

Previusly Izaya called her "Namie-san", and she called him "Izaya-san", but after six month together they become just "Namie" and "Izaya" for the other one.
They share a lot of secrets even about human's experiments and Izaya's plane, and also, they're like fiancés in particular moments that become important, as the dishes.

Namie starts to cook for him and the dishes become their own moments, that Izaya and Namie share happily.
One of the most adorable and meaning moment for them is when all the members of the chat eat hotpot in Mikado's apartment but Izaya isn't invited. So he asks to namie to cook hotpot for him and to eat it together with him.
They're a sort of family, and surely, they share the most solid bond that two dengerous persons like them are free to have.

The moment in which Shingen visits Izaya's apartment is one of the most interesting in order to know many particulars about Nebula's goals, but it's a moment in which Izaya thinks that he was going to kill Namie, because he was trying to intimidate her with a gun (but in a second moment we find that it's a toy) and Izaya was worried for her.

But their relation isn't only positive thing, they aren't a canonic couple because they're ready to betray the other one in every moments, as the ones in which Namie reveals to Kururi and Mairu that their brother is part of Dollard and he's kanra.

It's not clear how they feel for the other one, but surely more than they think.

Also, a great proof that they're close is the cover of the 7th novel.

Namie with izaya's jacket... isn't it proof of closeness?