Citing sources from Atletico, the Spanish publication adds that City are also offering €5m in bonuses.
In total, City could pay a total of €94m (£86.7m) for 25-year-old Gimenez if Atletico were to accept the offer.
But Atletico do not want to sell the Uruguayan and in response have referred City to his €120m release clause.
AS contacted Manchester City for confirmation but received no response.
City’s sporting director Txiki Begiristain has been negotiating with Atletico counterpart Andrea Berti and the club’s chief executive Miguel Angel Gil Marin.
Pep Guardiola wants a high-level centre-back to help City compete for the elusive Champions League.
Alternative to Koulibaly?
Although they have pointed City to Gimenez’s release clause, Atletico are reportedly thinking about the possibility of a sale.
The sale of Gimenez would offset the financial losses expected as a result of the Coronavirus pandemic.
City and Atletico did business last summer when Guardiola’s side paid the €70m release clause of midfielder Rodri.
Now they are back for Gimenez, who would receive a huge pay rise if he moved to the Etihad Stadium.
Atelti boss Diego Simeone will try to convince Gimenez to stay, something he couldn’t do when City came calling for Rodri.