Chelsea are hoping to tie N'Golo Kante, Jorginho and Mason Mount down to new deals, according to reports.
The Blues have made a fine start to the Premier League season, taking 13 points from the first 15 available.
The Reds had a quiet summer compared to some of their rivals as the club focused on handing new contracts to key members of Jurgen Klopp's squad.
Christensen is said to be close to penning an extension worth around £120,000 per week, but progress with Rudiger has been slower.
The Germany international is thought to be holding out for £200,000 per week, which is more than Chelsea are prepared to pay.
“I hope that they find solutions, but it's not only between coaches and players, and what we wish for,” Thomas Tuchel said of his compatriot's situation.
“And that's why it's between the club and the player’s agent at the moment.
“He's in the strongest league in Europe, in a big club, so not many reasons to change that. But in the end it's between him and the club, and that's in good hands.”
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