The centre-back has entered the final five months of his contract, and he is free to negotiate with overseas clubs.
Despite this, the Blues remain keen on extending his deal, but they have yet to make any breakthrough in negotiations.
Rudiger is on wages of £100,000-per-week. He previously turned down an offer of around £140,000-per-week.
The Blues have now upped their contract offer to almost double his current salary, but the 28-year-old has not changed his stance.
The west London giants remain optimistic of prolonging his stay, but don’t have much time with top European clubs lurking.
Paris Saint-Germain and Real Madrid have already opened talks with Rudiger’s agent over a possible pre-contract agreement.
Rudiger was close to leaving under ex-boss Frank Lampard, but the arrival of Thomas Tuchel in January 2021, changed his fortunes.
Tuchel instantly chose to play him on the left side of the back three, where he has put in several stand-out performances.
Meanwhile, there has been some talk that Rudiger may not suit the Blues when Tuchel finally opts to do away with the 3-4-3 system.
However, he has also answered those critics lately with some quality performances in a 4-2-2-2 set-up under the German tactician.
Rudiger is currently playing to the best of his abilities, and is widely considered as a favourite among a large section of the fan base.
The ball is firmly in the Blues’ court rather than Rudiger’s at the moment, and the onus will be on them to increase their salary package.
Rudiger will be aware that the next deal could the biggest of his career, and may hold up on making a decision until the summer.
He would earn a significant signing-on fee on a free transfer, and it remains to be seen whether the Blues can convince him to stay.