The Sun says City chief executive Garry Cook and football administrator Brian Marwood held talks with the Cameroon striker this week.
Barca have indicated they would be willing to sell Eto'o, 28, who has just 12 months left on his Nou Camp contract.
The fee is likely to be in the region of £20million but the deal now hinges on whether Eto'o fancies the switch to Eastlands.
City will offer him a salary comparable to the massive £160,000 a week earned by Robinho - but Inter Milan are also keen to sign him.
If City get the answer they want, it could be a busy weekend at Eastlands with a £15m move for Blackburn's Paraguayan striker Roque Santa Cruz, 27, also set to be finalised.