It has been suggested that Mojang might sell for more than $2bn (?1.2bn).
In June, the Swedish studio said it had sold about 54 million copies of its hit world-building game. Since then it has also been released for the Xbox One and PlayStation 4, further boosting sales.
Mojang's founder, Markus "Notch" Persson has previously publicly criticised Microsoft.
Ahead of the release of the Windows 8 operating system, he had claimed that the company was "trying to ruin the PC as an open platform" because it had wanted to certify third-party titles before their release.
He had also been a vocal supporter of independent studios, saying in 2012 that: "The more studios that can remove themselves from the publisher system, the more games that will be made out of love rather than for profit."Minecraft was released for the Xbox One a day after coming to the PlayStation 4
But, according to Bloomberg news agency, it was Mr Persson who "reached out" to Microsoft a few months ago.
It said that a source suggested that if the deal went through, Mr Persson would help with the transition, but was unlikely to stay on afterwards.
The New York Times also reported that a person briefed on the talks had said that Mr Persson was expected to leave within six months, but added that Mojang was keen to hold on to its younger developers.
A spokesman for Microsoft said: "We don't comment on rumour and speculation."
Mojang also declined to comment. Mr Persson did not respond to the BBC's questions.Virtual world
Minecraft was first released in 2011, and has since become the third bestselling game of all time and something of a cultural phenomenon.
The Victoria and Albert Museum in London has hosted an evening that explored its design culture, Lego has made spin-off toys, Ordnance Survey has created a version of mainland Great Britain within the game, and Denmark's government has also built a scale model of its country.Ordnance Survey used 22 billion Minecraft blocks to build a model of Great Britain
The title is set in a virtual world made of cubes of different materials, including dirt, rock and lava. Most of these can be used as building blocks and/or refined into usable raw materials.
Player use the blocks to build a shelter and to create weapons to defend their character against the game's zombies and other monsters.
They can also engage in huge construction projects to flex their creative skills.Lego - which served as an inspiration to Minecraft - has released tie-in products
One extreme recent example involved the creation of a working virtual hard drive.
Mojang has worked on other titles, but they have either fallen short of Minecraft's success - such as the role-playing game (RPG) Scrolls - or been shelved before completion - such as 0x10c, a space-themed game that Mr Persson had been working on until last year.'Worried gamers'
The Wall Street Journal was first to report negotiations between Mojang and Microsoft, saying that the deal could be finalised this week.
It noted that this would be the first multibillion acquisition by Microsoft's chief executive, Satya Nadella, since he took charge of the company.
Harry Bradshaw, who has interviewed Mr Persson on the Twitch video site, said that many gamers would be concerned by the news.A man has built a Minecraft model that acts as a hard drive that can store one kilobyte of data
"There have been cases in the past of big corporations taking someone else's game and turning it into a huge money-flip, just to see what they can make out of it," he said.
"Gamers will naturally be worried about this.
"And it does confuse me why Mojang would sell up, bearing in mind it is the bestselling PC game of all time and still sells thousands of copies a day."
One market watcher said that it was unlikely that Microsoft would pull existing Minecraft games from sale for the PlayStation, Mac, Android, Linux and other rival platforms.Mr Persson halted work on the game 0x10c in 2013 after complaining his code had been copied
However, he added that Microsoft's various systems might be given preference when it came to future products.
"Minecraft is one the leading IPs [intellectual properties] within the games space that is potentially acquirable," said Piers Harding-Rolls, lead games analyst at the consultancy IHS.
"The brand has the potential to continue for the next 10 years,
"So, as an investment - even one at a very high price - it still represents something of value to larger companies that are looking to acquire exclusivity around a highly-engaged franchise."