Anzhi sacked Gadzhiev with the club seventh in the Russian Premier League with five games remaining. Roberto Carlos and Gadzhiev's former assistant Andrei Gordeev have been placed in temporary charge.
The Dagestan-based club are bankrolled by billionaire Sulieman Kerimov and former Brazil full-back Roberto Carlos believes a big name will be targeted to fill the hot seat.
The 38-year-old, in remarks reported by the Daily Mail, said: "I am having dinner with the president and we'll speak about signings and coaches for next season.
"In terms of coaches, there are several names, such as Capello, Hiddink, (Wanderley) Luxemburgo and (Luis Felipe) Scolari."
Kerimov has already spent vast amounts on the playing squad with Samuel Eto'o the most prominent arrival, the Cameroon striker reportedly becoming the highest-paid player in the world following his switch from Inter Milan last month.
Balazs Dzsudzsak and Yury Zhirkov have also been recruited and Roberto Carlos is intending to discuss more players with Kerimov.
He added: "We will talk about names like (Frank) Lampard, (Steven) Gerrard and (Robert) Pires."