Last week, the Blues informally offered the defender a new deal worth closer to the £200,000-per-week.
The improved offer, however, still falls short of what Rudiger is asking for to stay at Stamford Bridge.
A bid even closer to that £200,00-per-week demand has now been offered to the 28-year-old and his representatives, with the club desperate not to lose too many defenders on free transfers this summer.
Club captain Cesar Azpilicueta and Andreas Christensen are also out of contract in June, with the duo linked with moves to Barcelona.
Meanwhile, Real Madrid and Paris Saint-Germain have offered Rudiger pre-contracts ahead of the summer transfer window.
Rudiger knows he can earn as much as £400,000-per-week – inclusive of signing on fees and bonuses – if he leaves Chelsea on a Bosman transfer.