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According to Italian website, Calciomercato, the World Cup-winning defender will drop into the free agent pool this summer if no fresh terms are agreed at Santiago Bernabeu.
Real had offered Ramos a new contract, but he has not accepted it.
English Premier League (EPL) challengers, Manchester United are now thought to be interested in offering the 34-year-old a lucrative two-year contract.
Imagine thinking Sergio Ramos doesn't love Real Madrid.— ???✵ (@ModricEle) January 6, 2021
He's our captain, he won't leave us.pic.twitter.com/Yz2JyP0oEM
In a piece of related news, Manchester United can go top of the EPL Table on Tuesday night.
Just a point against Burnley at Turf Moor will enable Ole Gunnar Solskjaer's team to go into Sunday's clash against Liverpool at Anfield with the champions trailing in their wake, raising hope within Old Trafford of a genuine title challenge this season.
Edinson Cavani is back available for selection after serving a three-match ban.