In the MCU, they know each other since the first movie. As Tom Hiddleston (Loki) said in some interviews, they share an un-spoken relationship because Loki knows it was Jane who -apparently- changed Thor for better, and he threatened Thor saying "Perhaps I'll pay her a visit myself", implying a twisted interest in her because of her relationship with Thor. Later, in "Avengers" movie, as Loki's invasions happens, she's hidden by S.H.I.E.L.D. because she was Loki's possible target, in the same way Erik Selvig (Jane's tutor) was brainwashed by Loki's sceptre. Eventually, they both meet in "Thor-2" events, where Jane slaps Loki and Loki claims "I like her"; however, after this event, their relationship is hardly developed due the amount of cut off scenes in the final movie montage. Loki, however, saves Jane's life twice.
This couple is totally fanon since neither in comics and moves they share any kind of romantic relationship. In MCU, this couple is supported because of several reasons. Firstly, Jane has no real interest in Thor because they don't truly know each other well, nor are able to keep a long term relationship; second, many fans seek to pair Loki with somebody and Jane happens to be one of the short number of female characters available; third, Loki and Jane have many common traits. Jane is a scientist, seen as a nerd and is usually mocked because of her beliefs, quite like how Loki is viewed by fans.
Jane also is a caring person, which is precisely what Loki needs according with the fandom. This pairing is hardly the most popular in the fandom due the fact Jane Foster isn't a widely loved character by fans, who usually see her as a dumb and poorly developed character. Loki, however, is very popular and many fans prefer to pair him with more popular characters. However, due the later movie events, this shipping base is becoming bigger.
|SHIPS||het||BobbiFitz • Brutasha • Clintasha • FitzSimmons • Fosterson • Huntingbird • Iron Widow • JessLuke • Kitscliffe • MayWard • NebulaRhodey • PeggySous • Philinda • Romanogers • Scarlet America • Spider Witch • Static Quake • Steggy • Winter Widow • Thunderwar • YoYoMack|
|slash||Capsicoul • FitzHunter • FitzMack • FitzWard • Hawksilver • Iron Husbands • IronStrange • Iron Winter • JackDaniels • Roquill • SamBucky • SamSteve • Science Boyfriends • Spiderlad • Stackie • Stony • Stucky • T'Chalcon • Thunder Science • ThunderShield|
|femslash||Cartinelli • Danbeau • Scarlet Widow • Simmorse • Valcarol|
|non-binary||FrostHawk • FrostIron • Frostmaster • Lokane • Scarlet Vision • Valki|
|family||Domestic Avengers • Iron Dad|
|CHARACTERS||m/f||Bucky Barnes • Natasha Romanoff • Steve Rogers • Tony Stark|