A sluggish start to the season - Inter sit ninth in Serie A, 10 points adrift of leaders Juventus after 10 games - has seen Mazzarri's future become the centre of frenzied speculation.
Former owner Massimo Moratti quit his role as honorary president last month after reports that he was unsuccessful in his desire to see Mazzarri sacked.
But Inter will be under pressure to return to winning ways when the 11th-placed Verona head to Milan on Sunday.
"We played with great desire [against Saint-Etienne]," Juan said.
"It's important that we play well against Hellas Verona, with this determination and desire.
"We're with Mazzarri. He's our coach."
And he has called on the support of the San Siro faithful for Sunday's clash.
"Will the support of the fans be important on Sunday? We want to do well," he said.
"[We] would like the fans to spur us on because my lads deserve it."
Verona lost 3-1 at home to Inter's city rivals Milan before being thumped 6-2 at Napoli last month.
Osvaldo returned from a thigh injury to feature off the bench at Saint-Etienne, but Yuto Nagatomo, Jonathan and Fredy Guarin are still battling problems.
Inter boast a good record against Verona, though, having won the last six matches between the teams in all competitions.
