La Liga giants Real have shown interest in Courtois since last year and the goalkeeper has fuelled speculation over a move by stalling over signing a new deal.
The Belgium international, who signed for Chelsea from Genk in 2011, will have just one year remaining on his current deal come the end of the season.
Courtois has admitted he plans to put contract talks on hold until the summer but Chelsea are reluctant to wait and have already discussed improved terms for a new deal, according to the Daily Mirror.
Courtois, 25, has previously spoken of his desire to return to Spain but it is claimed reigning Premier League champions Chelsea believe he will stay.
In 20 games of the 2017-18 campaign, Courtois has registered ten Premier League clean sheets. Chelsea are currently third, 16 points behind Manchester City and just one below United.
The Blues beat Brighton 2-0 on Boxing Day and manager Antonio Conte pointed to a lack of luck for his side falling behind City in the title race.
‘This season is different to last season,’ the former Italy boss said. ‘Last season a lot of times we won games and were a bit lucky – like against West Brom here, Sunderland.
Chelsea 2017/18 Fixtures
Stoke City (H) – Sat Dec 30
Arsenal (A) – Wed Jan 3
Norwich City (A) – Sat Jan 6
Arsenal (A) – Wed Jan 10 (EFL Cup)
Leicester (H) – Sat Jan 13
Brighton (A) – Sat Jan 20
Arsenal (A) – Wed Jan 24 (EFL Cup)
Bournemouth (H) – Wed Jan 31
Watford (A) – Mon Jan 5
West Brom (H) – Mon Feb 12
Barcelona (H) – Tues Feb 20
Manchester United (A) – Sun Feb 25
‘This season sometimes we could have more luck. We have great respect for Manchester City because they are doing an extraordinary thing.’
Chelsea face mid-table Stoke City on Saturday.
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