Spurs saw a £10.6million bid rejected last month but are understood to have upped their offer to £12m, says the Daily Mail.
Negotiations are continuing between the clubs but Inter president Giovanni Luigi has admitted he would prefer to sell Damiao in January - after the Brazilian season ends.
Luigi said: "We currently have three proposals for him - very good proposals.
"We know that the summer window in Europe values players more highly than in winter.
"It is important that we negotiate the best price for Damio. But we want to make a deal similar to the one with Sandro, and ensure that Damiao stays until the end of the current Brasileirão campaign. That is very important for our negotiations."