The Sun says Benitez and his head of recruitment Steve Nickson have been working hard to secure the services of the Paraguay international forward.
Nickson has been in America to speak to Almiron's agent and Atlanta.
And he and Benitez believe they have been successful in persuading their top January target to join the Toon and get a provisional deal in place.
A loan until the end of the season with the option to buy Almiron for £15million has been discussed.
But any move still needs the green light from Newcastle owner Mike Ashley.