Play the NPC. Don't play a "romantic NPC." Just have the characters talk about things they would logically talk about. If the player is actively seeking out the NPC to build their relationship, then you're doing it right.
It means getting into the NPC. It means coming up with their hopes and dreams and little anecdotes about their life. I've always found it helps if the NPC has some kind of tragic secret or something dark in their life, but that's far from a true requirement.
I suggest this. Open up a word processor, and start typing about the NPC. If you can't type up at least a full single spaced 12 point font page about the NPC, then they're not interesting enough for a player to want to engage with them on a long term basis.
I recommend this. Sure we can as GM and player sit down and negotiate the terms of attraction and the goals of relationship. But part of the fun of romance is its organic, uncontrolled nature. It also saves you from a lot of prep work.
Grab a 3x5" index card, scribble known details about the NPC (name, looks, 3 adjective personality...) and then separate a section for each tasks/routine, goals, and relationships. On the reverse, save for current news, quests, & secrets. Update accordingly, get in its headspace, and let loose.
i.e.:
Carlotta Durielwild, curly-haired brunette with fiery black eyes and commanding alto voice.
personality: passionate, quick-tempered, brooding.
job & routine: gunsmith, store open from M-F 10am to 6pm.
goal: match the quality workmanship of her grandfather.
relationships: Guillarme, silversmith --, Marquis Lac -, Duphrain, sheriff /, Lady Helene +, Armand, famed duelist ++. PC +.
(reverse)
news: mahogany gun handle shipment delayed.
rumors: Marquis Lac deliberately funds bandits to disrupt this town.
quests: a) find another source of good hardwood, fast! b) find evidence for truth in Marquis Lac and bandits rumor.
secrets: shot her last lover in the heat of jealous passion. she visits a lovely grave in the next county annually because of remorse.
Write it in pencil, edit regularly when they encounter each other, and let it flow organically. At worst, when the romance peters out or fails, you have a fleshed NPC and contact source.