The 28-year-old was transfer-listed by Barcelona last season but has found his best form under new boss Pep Guardiola.
Eto'o has given no indication he would be willing to leave the Catalan giants.
This is the type of player you should go for. Eto'o is more than world class The Ghost of Sonny Anderson In February, he said he wanted to stay at Barcelona beyond the end of his contract in 2010.
"Of course I want to continue, I'm very happy and see no reason to leave," he told TV station Canal Catala.
"I'm going to stay to the end of my contract and if the club want me, I would like to stay longer."
Manchester City certainly have the ambition and financial clout to lure the likes of Eto'o since their takeover last September by the Abu Dhabi United Group, backed by Sheikh Mansour Bin Zayed Al Nahyan.
But the club will not figure in next season's Champions League and are by no means guaranteed a place in the new Europa League.
emp_load.getEmpEmbeddedParams("emp_8028516");Officials are also likely to tread carefully in any fresh transfer bids for big-name players following the acrimony over their failed move for AC Milan's Kaka.
Manchester City's executive chairman Garry Cook was criticised after accusing Milan of "bottling" a £100m deal for the Brazilian star.
City manager Mark Hughes was at pains to play down the significance of Monday's meeting when asked about it at a news conference on Thursday.
"The aim of it was to build a relationship [between the two clubs] at all levels, it wasn't about individual players," he said.