Anzhi, who recently installed Andrei Gordeyev and Roberto Carlos as joint caretaker managers, have already made significant signings with Yuri Zhirkov joining from Chelsea and Samuel Eto'o signing from Inter but are intent upon becoming an elite club.
Tevez appears certain to leave Manchester City in January as an investigation into his apparent refusal to face Bayern Munich continues and, with few clubs both willing and able to meet his asking price and wage demands, Anzhi would appear to be well placed.
German Tkachenko, a star player agent, said he intended pestering City until they agreed to the sale - "Abramovich told us to go away ten times before agreeing to sell Zhirkov," he said - while a source at the club told the Tvoi Den daily: "The conditions for signing Tevez are very good."
Roberto Carlos has also spoken publicly about the club's desire to sign Neymar from Santos, as well as less feasible future moves for Cristiano Ronaldo and Jose Mourinho.
Real Madrid and Barcelona are understood to be leading the race for Neymar, with Chelsea having refused to meet his release clause, but it appears the Brazil starlet may yet consider a move to Anzhi if he fails to sign for either Spanish club.
"I would love to sign Cristiano Ronaldo," Roberto Carlos told Punto Pelota. "I have spoken to Santos' Neymar and if Madrid or Barcelona fail to sign him then we will.
"The objective is also to sign Mourinho but not only him. We have tried to speak with various experienced coaches who are well known. We want to make Anzhi like Real Madrid and Barcelona and I say this with intent."
He added that Eto'o is enjoying his time in Russia. "He has always been a 'Madridista'. He is happy and not just because of the money, but because of the fans who have been treating him well."
Anzhi are currently eighth in the table despite making a strong start to the season and dismissed long-serving coach Gadzhi Gadzhiyev at the end of September.