Anzhi is expected to give Eto'o a three-year contract worth ?20 million ($29 million) per season, eclipsing the estimated ?12 million ($17.4 million) that Cristiano Ronaldo earns at Real Madrid and the ?10.5 million ($15.2 million) that Lionel Messi is paid by Barcelona.
Inter was negotiating with Anzhi officials in Milan for a transfer fee between ?20 and ?30 million ($29 and $43 million) for the 30-year-old Eto'o.
''We're very close. We could close today, tomorrow or next week,'' Anzhi representative German Tkachenko said late Wednesday, according to the ANSA news agency.
Eto'o also met with Anzhi officials in Milan on Wednesday.
Anzhi, from the troubled southern Russian province of Dagestan, is owned by billionaire oil tycoon Suleiman Kerimov.
In Forbes magazine's latest list of the world's richest people, Kerimov's net worth was estimated at $7.8 billion, tying him for 118th place with Silvio Berlusconi and family.
Berlusconi, Italy's premier, is also president of AC Milan.
Anzhi's captain is 38-year-old Roberto Carlos, once the standout left back for Brazil.
The transfer fee would also make it one of the most expensive deals this season, near the ?29 million ($42 million) Barcelona paid Arsenal on Monday to bring Cesc Fabregas back to his boyhood club.
The biggest deal during this transfer window was the 38 million pounds ($62 million) Manchester City paid Atletico Madrid for Sergio Aguero last month. Also, Paris Saint-Germain paid ?42 million ($60 million) to Palermo for Javier Pastore.
''When an offer like that comes around, it's difficult to match or refuse,'' Inter president Massimo Moratti told Wednesday's Gazzetta dello Sport.
Moratti added that UEFA's new financial rules have forced clubs to evaluate their budgets and spending more closely. This season's club accounts are the first ones that formally have to be given to UEFA for inspection - by June 2013.
''Financial fair play is serious,'' Moratti said. ''Whoever doesn't respect the parameters will find themselves excluded from European competition.''
Inter is eyeing Diego Forlan of Atletico Madrid, Ezequiel Lavezzi of Napoli or Carlos Tevez of Manchester City as a replacement for Eto'o. Inter's other forwards are Giampaolo Pazzini, Diego Milito and Goran Pandev.
Meanwhile, playmaker Wesley Sneijder appears set to remain at Inter.
Eto'o led Inter with 37 goals in all competitions last season, having replaced Milito as the club's biggest scoring threat. He was a key member of the Inter side that won the Champions League in 2010, often accepting an attacking midfield or defensive role for then-Inter coach Jose Mourinho.
Before joining Inter in 2009, Eto'o played five seasons with Barcelona, helping the Catalan club to two Champions League titles.
Makhachkala is plagued by near daily violence from insurgency that has spread across the North Caucasus after two separatist wars in Dagestan's neighboring region of Chechnya.
Anzhi players live and train at a training camp outside Moscow and travel to Makhachkala for home games, flying about 2,000 kilometers (1,250 miles) 15 times a season.
Kerimov's worth was nearly the size of Dagestan's gross regional product in 2008.
A Dagestani native, Kerimov has been pumping money into the team ever since he bought it in January, with some critics comparing the tycoon's spending spree to a ''feast in time of plague.''