The big-spending Russian side are reportedly on the verge of offering £33m for the Serbian international, and the Old Trafford club are thought to be open to offers.
The 30-year-old centre-back has struggled with injuries in the past 12 months, and is currently sidelined with his second serious knee injury in the past year.
Anzhi are believed to be ready to offer Vidic a mammoth £200,000-a-week wage to tempt the Serbia star to Russia. Anzhi already pay former-Inter Milan striker Samuel Eto'o the highest wage in football - the Cameroonian forward reportedly earns close to £300,000-a-week.
Vidic spent two years in Russia with Spartak Moscow, and Anzhi are confident they can put a package together attractive enough to get the Serbian's attention.
The former-Spartak star joined Manchester United for £7m in 2006, and has since gone on to become one of Sir Alex Ferguson's most important players.
Vidic has made more than 171 Premier League appearances for Manchester United, but the 30-year-old missed the vast majority of last season's campaign after damaging cruciate ligaments during Manchester United's Champions League defeat to Basel.
The Serbia international returned to the heart of United's defence this season, but looked off the pace on his comeback, before damaging the same knee in the warm-up to his side's victory over Liverpool in September.
The Manchester United defender underwent surgery, and faces up to eight weeks out. With Phil Jones, Chris Smalling, Jonny Evans and Rio Ferdinand in the same position, Ferguson could be tempted by a mega-money offer from the Russian club.