Sky Sport Italia and Gazzetta dello Sport report the Nerazzurri have opened talks with representatives of the former Real Madrid man.
The 27-year-old joined City in the summer of 2017 for €30m, and the Premier League side will want to recuperate as much of that fee as they can.
Therefore, an agreement between the clubs could be found in the region of €20-22m for the Brazilian, who still has three years left on his contract.
Indeed, La Gazzetta claims he has already agreed a four-year contract with Inter worth €4m a season after tax.
Danilo has recently found himself down Pep Guardiola’s pecking order, making just 20 appearances this term.
Primarily a right-back, he can also fill in on the left side of defence.