The Saints have followed German side Wolfsburg in making a written offer for Uruguay international Nunez.
And it is reported that Southampton have a ‘slight advantage’ in their head-to-head with Wolfsburg for Nunez.
Nunez, 21, scored 16 goals in 30 games in Spain’s second tier this season and is also on Roma and Napoli’s radar.
He has been compared to compatriot Edinson Cavani and interest is growing across Europe.
Almeria’s failure to win promotion to La Liga means it is almost certain that Nunez will leave.
The Andalusian club lost to Girona in their promotion playoff on Sunday, with Nunez playing the full 90 minutes.