The Sun says the Russians want to follow their £20m capture of Inter striker Samuel Eto'o by adding a top-class defender.
And United skipper Vidic, 29, is at No 1 on their wanted list — with a top Russian source telling The Sun that a bid for the Serbian is "imminent".
The news will have United boss Alex Ferguson sweating, especially with just two days left of the transfer window.
Anzhi are controlled by Russian billionaire Suleyman Kerimou, who has an estimated personal fortune of £5bn.
He stunned the football world last week with his swoop for Eto'o, who is said to be on a whopping £300,000 a week.