The only person with enough canonical evidence to develop a possible romantic relationship with Neku is his first partner, Shiki. Though it is unknown if Neku actually does take Shiki on a date after they return to the RG, I'll recap the most memorable moment in the game that they shared: End of Week 1, after defeating Game Master Higashizawa.

Neku: Game over.
Shiki: That's it?
Neku: We did it! We won!
Shiki: Woo-hoo! Now we'll come back to life! !? Wh-what's going on?
Neku: !? Shiki?
Shiki: Huh? I don't think this light is bad.
Neku: Yeah... Feels almost like a...pat on the back.
Shiki: A warm welcome back to the world of living! And you said "Shiki"! Neku... That's the first time you used my name.
Neku: Oh really?
Shiki: Neku? See you on the other side. You know the meeting place. Hachiko!
Neku: Heh. It's a date.

Soon after this, it is discovered that only one person can return back to the RG. Neku is stuck back in the Reapers' Game for another week, but this time with Shiki as his entry fee. It is unknown that since this happen Neku actually harbors feelings for Shiki, but one can make the safe assumption that he greatly values their friendship and their time together during the first week.