What is Izaya’s relationship with Namie? Can it improve?
In book 3, which I'm currently reading, Izaya and Namie are not yet very close, but they are in the process of building real trust towards one another.
Namie is still scared of Izaya, and she's not wrong to be! Izaya is a very powerful individual at a very young age. He has the capacity to do things most people wouldn't, and probably shouldn't. Her life is literally in his hands; he knows everything about her and her situation, and her greatest asset and trump card (Celty's head) is in Izaya's possession.
But Izaya never abuses that power. He keeps his promise perfectly; he arranges hotels and eventually whole apartments for Namie, keeps her moving and off anyone's radar, and pays her incredibly, ridiculously well even after all that. Thus Izaya proves to her that he's reliable and remarkably trust worthy... So long as you're on his side.
On top of that she'll eventually start seeing more and more of who he actually is. And Izaya is a lot of things, but the side Namie gets to see, working so closely with him is very different from the seemingly cold and careless asshole act that a lot of the other characters see. She gets to know a guy who's defined mainly by his intelligence, who is full of childish curiosity and excitement, and who is deeply hurt by the rejection he faces from just about everyone.
Jealousy and anger aren't good traits, but Namie finds herself in a position from which she can see their cause; and she can understand it. At first she refuses to feel sorry for him, still convinced he deserves it. She never really gets past that opinion; after all Izaya did make his own choices leading to this situation; but she can see the rift between who he is and what he pretends to be. Namie is one of the few characters (the others being Mikage, possibly Kine and Shinra) who actually understands that Izaya genuinely cares for the people he plays with. Between that and the slow transition from fear into respect, she comes to care for him, even if she'd probably never admit it out loud.
On Izaya's side, something similar happens. From the start, he sees Namie as a highly skilled, intelligent and capable woman, with her love for her brother being her one big flaw (which he does find pretty creepy). He probably figures such a flaw makes her easy to manipulate, but in reality, all he actually does with it is tease her. He seems to actually hold out on info regarding her brother, almost as if he's trying to encourage her to let go of that madness. But on the other hand, he can't help but admire her single minded devotion when she doesn't, and continues to chase her little brother.
At the same time she proves to be an invaluable asset to him. Namie is capable, professional and adaptable, she can do her job without letting feelings get in the way, and she becomes somebody that Izaya can actively rely on. Which is not something he's used to, and it certainly puts her in a special position, especially combined with the sheer amount of time they spend together. Especially when she starts occasionally giving in to his more domestic demands, I can't help but think she's found a special place in his little heart. Izaya isn't used to being cared for, and sometimes... That's just what Namie does.
Unfortunately Namie does in the end get caught up in a mess as she goes after Mika, leaving Izaya behind, but this may have been planned or agreed to, because Izaya does not appear to feel betrayed. In fact Namie is one of the extremely few people that he actually contacts after his disappearance, and tells about his work in Kanto; which implies both an incredible amount of trust (by his standards) and a strong bond.
I think at the end of the series, Namie is Izaya's real best friend, despite him still only naming Shinra as such.
I'd say there's absolutely a possibility that they were more than friends, but even then I can't see their feelings as being especially romantic in nature. They're definitely not in love, but I can see them in a relationship of convenience and comfort.