Although no romantic undertones are evident in either the manga or anime, and their relationship skirts around the term "friendship", there is a firm sense of respect from Eren directed towards Levi built up over the course of the manga. Levi cares about Erens well being of course, but that is just apart of his job assigned to him by Commander Erwin. On Eren's side, Captain Levi of the Survey Corps represents everything he's come to admire and respect: freedom from the claustrophobic outer walls, hatred of titans and a desire to avoid the fate set out for him when Eren inquires about Levi's past, Petra informs him that he was previously a thug in the underground city. The start of their relationship in canon is thus dominated by Eren's hero-worship, and Levi's belief that Eren is little more than a monster with the potential to become an asset to humanity's cause. After being taken into custody, Levi briefly asks Eren, "What is it you want to do? Eren's attitude towards titans interests Levi enough that he decides to take responsibility for him, vowing that if he "betrays us or goes berserk, I'll kill him on the spot.
LEVI & EREN FANART
Levi Ackerman/Eren Yeager - Works | Archive of Our Own
Eren x Levi
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Two of the series most popular male characters, Eren for his obsessive resolve and Levi for his rugged good looks, have a relationship that has peaked some considerable interest— or, at least, their lack of a relationship has. However, there are times when there seems to be some form of a friendship between these two. Being both skilled fighters, physically and tactfully , these two have worked with one another for quite some time against the Titans that threaten humanity. Their similarities help to gain some mutual grounds between them, regardless of their difference in rank and oftentimes bring them together or forces them to trust one another when otherwise they would clash. Through analysis and fans theories, there have been many speculations on if these two have a friendship or not.