Chelsea manager Thomas Tuchel revealed after Monday's (AEST) late win over West Ham that Rudiger intends to leave the club when his contract expires at the end of the season.
Tuchel suggested that Rudiger's decision was influenced by the financial restrictions imposed on Chelsea while a new owner to succeed Roman Abramovich is being sought.
According to widespread reports on Tuesday, Madrid have had no such problem agreeing terms with Rudiger and an announcement is expected to be made imminently.
The Germany international has spent five seasons at Stamford Bridge and Jorginho admits his team-mate's absence will be felt both on the pitch and off it.
"He's been here a long time so if he leaves then we're going to miss him," he told Sky Sports News. "He's a big personality and he's helped us a lot.
"We've had amazing times here; he's a good friend of mine and of course if he leaves everyone at the club will miss him, not just me."
Asked what he will miss the most about Rudiger, Jorginho said: "His craziness. He makes me laugh a lot; all of his jokes and laughs we've had together – that's the nice part."
Rudiger has started 47 games this season, which is five more than any other Chelsea player, with Jorginho sixth on that list (42 appearances).
Indeed, no player from a club in Europe's top five leagues has started more matches in 2021-2022 when taking all competitions into account.
Rudiger played no part in Chelsea's victory over West Ham last time out, however, a game in which Jorginho missed a penalty before Christian Pulisic snatched a late victory.
Jorginho was criticised for his spot-kick style after sending his effort straight at goalkeeper Lukasz Fabianski, but the midfielder suggested VAR was partly to blame for the miss.
"This sort of pressure, especially in games like this when you need to win and it's last minute, you try to isolate your mind from all that pressure even though it's hard," he said.
"Then you have the VAR so it takes longer, it's harder to keep focus.
"What you need to do and what I try to do is I try to isolate my mind of all this pressure, and unfortunately this time it didn't work."
Prior to his penalty miss against West Ham, Jorginho had converted each of his last 13 for Chelsea (excluding shoot-outs) in a run stretching back to Boxing Day 2020.
"When you have that little moment of doubt that's usually when you make the mistake," the Italy international added. "Maybe I had too much time to think about it, I don't know.
"This time it went like that and I'm sorry. I feel bad because it's not a nice feeling."