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.
And just when it looked like the team had turned the corner, after scrappy wins over Cesena and Sampdoria, Inter lost 2-0 at then-bottom Parma last weekend.
A 1-1 UEFA Europa League draw at Saint-Etienne followed on Thursday - an impressive result that kept Inter four points clear at the top of Group F.
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."
Mazzarri took over at Inter last year after a successful four-year stint in charge of Napoli.
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.
Draws - both 1-1 - have followed against Lazio and Cesena, with Verona going in search of their first win in more than a month on Sunday.
No Verona player has scored more than two league goals this term, while for Inter, Mauro Icardi and Pablo Daniel Osvaldo have weighed in with five and four respectively.
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.
That included a 4-2 triumph in the corresponding fixture last term, in which both sides had a man sent off.
