Initially, Namie and Izaya had very little contact outside of what was required for their jobs. Namie even expresses some controlled, but visible, annoyance with Izaya's personality. However, she does seem to have confidence in his abilities to the point to where Izaya is the first person she turns to after the incident with the Dollars.
Once Namie agrees to work as Izaya's secretary, she tries to keep a professional mood between them but it is constantly undermined due to her obvious disdain for her new employer, as well as Izaya's flippant personality. She follows Izaya's schemes with reluctant cooperation, but has shown a few moments of disobedience such as informing Horada about Mikado being the leader of the Dollars, and giving Kururi and Mairu the link to the chat-room. She has also shown some disturbance and even fear at what Izaya was capable of doing. Within the light Novel series, their relationship is a bit more fleshed out. From Izaya's point of view, he appears to be more or less indifferent towards Namie. He has stated that he views Namie as a horrible person and usually makes fun of her for her incestuous love for her brother. However, Izaya enjoys teasing people so these insults are likely a product of that enjoyment. Despite his generally low opinion of her, he states that he loves her in the same way he loves all other humans (although that isn't something she exactly takes comfort in).
From Namie's side, she views Izaya as an idiot/child and has even contemplated killing him on numerous occasions. Namie claims that the only reason she hasn't poisoned him yet is due to the large paychecks she receives. However, she has shown a surprising level of patience with Izaya and has frequently been shown going above and beyond what would normally be required of her such as when she agreed to let Akane Awakusu stay with her while she ran away from home.
Since Namie initially became his secretary, Izaya seemed to take pleasure in having watched Namie fall so far from grace. Shortly before she agreed to work for him, her family's company was bought by Nebula. Izaya dangles this fact over her head, attempting to get to her though it seldom works. Due to Namie essentially being on the run from Nebula, Izaya provides Namie with an apartment to live in, moving her around frequently to protect her. Although Izaya himself owns multiple safe-houses, Namie nonetheless seems to appreciate Izaya's protection even if she dislikes him. The more she works with him, the more she finds out about his personal secrets.
For example, Namie has discovered that Izaya is an incredibly lonely individual. This was shown when he found out that everyone else in the chat-room had been invited to Shinra's for hot pot and even Masaomi had Saki to eat with. Without realizing Namie had been reading the chat, Izaya asked her if she would allow him to treat her to some hot pot. Namie, realizing that he was only asking in order to comfort himself and his wounded pride, refused and poured salt in the wound by rubbing it in her employer's face that she had seen what he was up to and that he was only asking because he was lonely. She asked him if he would 'mind not using her to satisfy his vanity'.
In addition, Namie has discovered that Izaya greatly dislikes his sisters being let in on any of his business, Namie frequently tells them information that Izaya himself denies them. She also pays them the same rate she gets paid by Izaya to spy on Seiji for her rather than ask Izaya for information on Seiji himself, technically giving them Izaya's money to do his work. Upon realizing that Namie told his sisters about who Celty was and that she reads up on what happens in the chat-room, Izaya calls her 'a force to be reckoned with', suggesting his respect for her abilities even if he doesn't enjoy being scrutinized.
Izaya is often mentioned or shown to be irritated or uncomfortable with how well Namie does her job and how much she pays attention. In one instance, he gives her a great deal of work, only to seem perturbed when she finished it a very short while later. She even went above and beyond completing the tasks he gave as if knowing it would annoy him more. Despite his wariness of her knowledge, Izaya can't seem to help telling Namie information whenever she asks about something, such as the meaning of the code-word 'chocolate' referencing the yakuza's missing gun. While it's possible he might let her in on such pieces of knowledge simply to see what she'll do with them, it seems far more probable that his main reason for telling her things like his plans are because Izaya has no one else to gloat to.
Although both Izaya and Namie claim to dislike the other or find them annoying, Izaya still goes out of his way to save Namie when she's abducted. One of the suggested reasons for this is the wealth of knowledge Namie has from working with him, Izaya worried over that knowledge falling into the wrong hands.