Southampton boss Mauricio Pochettino confirmed his interest in Damiao on Thursday, while Spurs are known to be long-term admirers of the Brazil striker.
However, the Premier League duo now look set to miss out to Napoli, who 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.
The exact transfer fee for Damiao - said to be around £18million - remains unclear, with Prates adding: "The reported value is a lie."