Jurgen Klopp has been left short in defence following long-term injuries to Virgil van Dijk and Joe Gomez, while Trent Alexander-Arnold is also set for a spell on the sidelines.
Van Dijk and Gomez are in battles to play again this season with Joel Matip the only recognised senior centre-back at Anfield.
Jones is often mocked at Old Trafford having himself struggled with injuries and made plenty of mistakes when he does play.
And United fans are suggesting Liverpool sign Jones, with one suggesting on social media: “Phil Jones and Marcos Rojo are available little injury prone so would fit in well.”
While another added: “We’re willing to give you a defender right now for free? You can have Sir Phil Jones for free. We understand your situation & even though you’re our biggest rivals but we’re still willing to help you in these difficult times.”
And a third wrote: “Wonder If Liverpool Fancy Taking Phil Jones Off Us Now Gomez Is Out.”
Another simply commented: “They can have Phil Jones.”
With one more putting, in a similar vein: “Can send them Phil Jones on loan.”
Dayot Upamecano has been linked with a move to Anfield, though one fan wrote: “Serious question… is Upamecano better than a fit Phil Jones? Cuz I certainly don’t think so.”
On signing a new centre-back, Klopp had said in the summer: “The situation is obvious.
“We have a lot of centre-backs, pretty much all of them apart from the three we had last year are all pretty young, which is good.
Get the latest transfer news straight into your inbox!
Want to be on the ball with all of the latest football news?
Well then sign up for the brilliant new Daily Star Sport email newsletter!
From the latest transfer news to the agenda-setting stories, get it all in your email inbox.
How do you sign up?
It only takes a matter of seconds.
Simply type your email address into the box at the top of this article and hit 'subscribe'.
And that's it, job done. You'll receive an email with all of the top news stories every single morning.
You can find out more information on our email newsletter on this link here.
“A lot of space for development and they all did well; Sepp did well, Ki-Jana prefers right-back but did well, Billy did really well considering his age especially, and Nat Phillips came back from Stuttgart full of confidence.
“But Fabinho can play it as well, he did it against Bayern in the Champions League once. It worked pretty well because he is a smart player.”