Simon Jones of the MailOnline reported the Bundesliga club—who currently loan the player from the Blues—are pushing to make the deal permanent after Christensen impressed, making his Champions League debut for the German team.
However, James Whaling of the Mirror said the Reds are "eyeing a deal" for the 19-year-old, as they look to swell their ranks with talented youth.
Here is a video of the defender's recent form for his loan club:
Monchengladbach sporting director Max Eberl confirmed his club want to bring the player to Germany on a permanent basis, per Jones.
"Our intention is to try and get into talks with Chelsea in the summer," Eberl said. "Andreas fulfils everything we've seen in him. And we are delighted that we were able to take him on loan without any ifs and buts."
BT Sport's Archie Rhind-Tutt recently tipped his hat to the youngster:
Despite Eberl's comments, Chelsea could now recall Christensen to Stamford Bridge after the long-term injury to Kurt Zouma.
The Telegraph's Matt Law claimed the Blues might take the defender back 12 months early, as John Terry's future remains unclear. Christensen has publicly stated he is happy in Germany, saying: "I will give it my best in the two years here [at Monchengladbach] and see what the future has in store for me. It's of no use to return [to Chelsea] if I don't get any playing time there. A lot can happen."
In other news, Liverpool's hopes of signing William Carvalho appear to be dashed after the Portugal prodigy rubber-stamped a new deal with Sporting Lisbon.
The midfielder has signed with his club until 2020, according to Charlie Phillippe of the Express. However, Carvalho's new deal includes a £35 million buyout clause, allowing a big-spending team to swoop for the player.
Christensen would be the perfect technical player for manager Jurgen Klopp to build his back four around, but it is his potential that would make him a valuable capture.
His performances in Germany have not gone unnoticed by his parent club, who themselves need to rebuild after a shocking season.
The defender's price is likely to rocket if his current performance value is maintained, and Europe's elite clubs would see him as a tidy investment for the near future.