The Brazil striker is expected to be the replacement for Edinson Cavani at Stadio San Paolo.
One of Damiao's representatives has admitted that there is a clear front-runner in the race to land the 23-year-old.
Vinicius Prates told Radio Gaucha: "There is a club that intends to sign the player. The interest exists."
Damiao has emerged in the last week as being one of the names on the top of Mauricio Pochettino's shortlist of potential signings.
Tottenham had been long-term suitors but their interest is understood to have cooled in recent months.
And despite Saints' attempts, new Napoli boss Rafael Benitez is reported to be the favourite to bring Damiao to Europe.
However, the exact transfer fee - said to be around £18million - remains unclear, with Prates adding: "The reported value is a lie."