The Journal reports that if such a deal goes through, it would be valued at more than $2 billion and "could be signed as early as this week."
We have reached out to both Microsoft and Mojang for comment.
As of June 2014, the independently-made world-building sandbox game sold almost 54 million copies across all its available platforms, which include Android, iOS, PlayStation 3, Mac, Windows PC and Xbox 360. The game launched on the PlayStation 4 last week.
Mojang officially released Minecraft in 2011 after a two-year beta period. The game is the studio's only major release to date. The development team has created mini-games like Catacomb Snatch and Battle Frogs as part of game jams. The studio is currently developing a tactical role-playing game called Scrolls.