The 85,454-seat venue is currently named after its former president Santiago Bernabeu.
Speaking on Spanish television, the president of Microsoft in Spain said discussions were at an early stage.
“They’ve raised… renaming the stadium and we’re only talking about this possibility,” said Maria Garana.
Online reaction to the news included possible new names for the stadium.
“At most the stadium will be called the Microsoft Estadio Santiago Bernabeu,” said one forum member.
“The Gates Ground?” was a suggestion on Twitter by @HuffPostUK, in a nod to the company’s founder, Bill Gates.
Others wondered whether Microsoft would use the opportunity to promote its products.
“Will it be called the Surface and feature players like Cristiano RonalDOS?” tweeted @tomwarren from technology news website The Verge.
The Surface is Microsoft’s tablet computer and DOS was one of the company’s former computer operating systems.
The announcement was not well received by all fans of the club.
“I’m a Real Madrid fan, I hate Microsoft forever if they touch anything related to Madrid,” said Haidi Farnoud.
If Microsoft are successful in obtaining the naming rights, they will join a group of several other companies whose names are linked to stadiums.
In the English Premier League, the airline Emirates has a tie-in with Arsenal, while Etihad sponsors Manchester City’s ground. The German club Bayern Munich play their home matches in the Allianz Arena after signing a 30-year deal with the financial services company.
But not all stadium renaming deals have been successful. When Newcastle United changed the name of their ground from St James’ Park to the Sports Direct Arena in November 2011, the local council refused to use the new name on official street signs.
At the time, the council said the decision to change the name would “upset the overwhelming majority of fans”.