This latest episodes raises serious doubts over Lozano's future at Napoli following an underwhelming campaign which has yielded just three goals.
It has been a difficult campaign at Napoli, with Allan having been sent from training following a similar situation in February, according to ESPN sources.
"Anyone who is tired, who doesn't feel like it, who isn't sharp mentally, as far as I am concerned, he can just stay in the locker room and miss a day of training," Gattuso told RAI later on Monday.
"Everyone knows that if they step onto the field with me, they have to go at full speed. I won't allow anyone to ruin a training session."
Napoli began the campaign as expected title challengers with Carlo Ancelotti as manager. However, a shocking run of domestic results, combined with a player mutiny as they refused to go on a training retreat, resulted in Ancelotti's sacking, despite reaching the Champions League round of 16.
Gattuso was appointed in December and, initially, did not see an immediate upturn in form with four defeats from his first five matches. Things improved before the league was stopped due to the coronavirus outbreak with five wins from their next six matches -- including a 2-1 success over leaders Juventus.
As well as being in the Coppa Italia final, Napoli still have an outside of chance of making the top four as they sit nine points behind Atalanta, who have played a game fewer.
In the Champions League, they have their round-of-16 second leg with Barcelona having drawn 1-1 at the San Paolo.
Information from ESPN's Eduardo Fernandez-Abascal was used in this report