Eto'o appears to have been made available for transfer by Anzhi after the Russian side revealed planned "changes to the budget parameters", fuelling speculation some of the club's big earners could be offloaded.
The Cameroon international was part of Mourinho's Inter Milan side that won the Treble in 2009/10, playing a key role as the Italian giants clinched the Serie A title, the Coppa Italia and the UEFA Champions League.
The 32-year-old scored 16 goals for Inter in that campaign - before leaving for Anzhi in 2011 - and is eager to link up with Mourinho again at Stamford Bridge.
"There is only one Jose Mourinho. I have played under a lot of great coaches, but there is no character in the game like him," Eto'o said.
"He is one of the best coaches in the world, and we had a lot of success together at Inter Milan.
"I would like to have the chance to play under him again because it is never boring when Jose is around."
Mourinho is keen to lure Manchester United striker Wayne Rooney to Chelsea and has had two bids rejected for the England striker.
However, Eto'o is willing to bolster Chelsea's forward line should their pursuit of Rooney prove unsuccessful.
"Chelsea are a great team, and when we have all the offers on the table I will discuss with my agent and see what the next step is," he added.