With Brienne and Jaime, Martin comes back to a romantic theme he uses with quite often in the series (most notably with regards to Sansa): that of Beauty and the Beast, only everything is tangled up so both Jaime (physically) and Brienne (morally) are beauty and reversed, they are both the beast. But I think romantic elements of the bond between Brienne and Jaime are very much intended by Martin: Beauty and the Beast don’t become best friends; they do learn to know each other as friends but they fall in love and they end as lovers. (The story of Cupid and Psyche, which has elements of Beauty and the Beast, is an allegory for the soul’s search for love, and you can tell me until you’re blue in the face that friendship is also a kind of love - and I agree! - but Psyche is actually sleeping with Cupid and is even pregnant when her family convinces her she’s married to a monster so she goes to kill him. So you know, it is about sexytimes too!)
I think Brienne having a physical attraction to Jaime and then being “friend-zoned” by him would be a kind of redundant repetition of her arc with Renly, and given how much Martin has emphasized her physical ugliness (complete with giving her a hideous facial scar) and how he seems very determined to show that beauty is skin-deep etc. (so Dany’s attraction to the handsome Daario is kind of laughable, but Gilly’s attraction to Sam is sweet and wholesome), I think it would be the culmination of that to have (beautiful) Jaime fall in love with (beastly) Brienne after he has become more beautiful (on the inside) thanks in part to her inner beauty.
I also keep coming back to Jaime’s dream: In a dream in which he is rejected by Cersei, his ideal of love, and accused by Ser Arthur Dayne, his ideal of Knighthood, the person who stands by him and defends him (as they are both naked) is Brienne, and this is how he sees her: “in this light she could be a beauty, in this light she could be a knight.” If we are meant to infer that Jaime and Brienne will be nothing more than friends, then there is no reason for this dream to note that Brienne is both a woman AND a knight or to juxtapose her with both Cersei AND Arthur Dayne. If we were simply to infer that Brienne is a someone Jaime considers a friend, then we might have gotten at most his juxtaposition of Brienne and Arthur Dayne, the example of the true meaning of chivalry that Jaime returns to over and over again. Instead, we also have Cersei, the only woman Jaime has loved until now and had a sexual relationship with, turn away from Jaime just before Brienne appears in Jaime's dream naked and "with more of woman's shape."
I’ve also been thinking about honor and how in patriarchal societies like most of Westeros (exceptions for Dorne and the Wildlings) a woman’s honor involved her sexual activities (a woman “dishonors” her husband by having an affair; a woman is “dishonored” by rape) and her sexual choices reflect on the man who “owns” her (husband, brother, father). Whereas a knight’s honor is separate from sexuality (mostly, unless he’s in the Night’s Watch or Kingsguard) and concerned with protecting the weak, defending the realm etc.
Brienne is one of the few (maybe the only?) characters who has both kinds of honor because she is a woman as well as a knight and both kinds of honor tie her to Jaime, who is one of the few people who readily accepts the dual nature of Brienne’s honor. In contrast to the cruel knights in Renly’s camp who bet on who will deflower her (and whose focus is on her “honor” as a woman) and to Randyll Tarly who literally tells her that if she’s raped it’s her own fault for not adhering to her prescribed role, Jaime saves Brienne from “dishonor” as a woman by his lie regarding sapphires because he understands how that will scar her on the inside. Later, he vouchsafes her honor as a knight to Loras and then goes even further in giving his own “last chance for honor” into her keeping. It’s such a gorgeous progression and it tells me that as much as Brienne knows Jaime’s sins and loves him anyway, he knows who she is on this level that no one else does. And I think a romance between them wouldn’t cheapen that knowledge and love but enhance it.