Chelsea defender Rudiger is a mainstay in Thomas Tuchel's side but his current deal expires in 2022.
United are one of a host of clubs interested in Rudiger and Rangnick has an ace up his sleeve.
The 63-year-old is said to be close with Rudiger's agent and half-brother Sahr Senesie according to Christian Falk of BILD.
Senesie would be willing to negotiate with United in an attempt to seal his half-brother a lucrative deal.
Thomas Tuchel is keen to keep Rudiger at Stamford Bridge given his form since his arrival in January.
But the Germany international is demanding more than the £160,000-per-week Chelsea are offering.
Rudiger instead wants a weekly wage of £230,000-£290,000-per-week.
Real Madrid, Bayern Munich and Paris Saint-Germain have all registered interested in the 28-year-old too.
He recently admitted a summer transfer was on the cards before swiftly changing his mind.
"Yeah [I was close to leaving], the way things were going for myself but also for the team," Rudiger told Sky Sports.
"But in the winter period I was not keen to leave, to be honest. It was more in the summer period where I wanted to leave.
"But Thomas Tuchel came and the rest is history. He opened a new door... he played a big role."
Rudiger has played 15 games in the Premier League this season, scoring twice to help Chelsea up to third.
The Blues also have Thiago Silva, Andreas Christensen and Cesar Azpilicueta entering the final six months of their contract.
Silva and Christensen are expected to re-sign while club captain Azpilicueta's future is still up in the air.
Tuchel is confident that all four men will renew their deals.
"We're on it with patience. We're on it also with confidence and we are on it with trust in the relationship between the players and the club," he said prior to Thursday's clash with Everton.
"I think all of these players know very well how much we appreciate them and the big role they play in our plans. I think they are very aware of it and will not throw it away like this.
"Of course it's their right now to do things, but it's also their right to stay with us and be a part of the future of Chelsea."
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