Ahead of Sunday's Serie A match at Bologna, Napoli are languishing in sixth place in the standings.
Following two wins a draw and a defeat, they trail leaders Inter Milan by five points.
The results have not met the ambitions of the club, marking a challenging beginning for Garcia who took charge of the club in June following the departure of Luciano Spalletti.
"Every time I meet with fans in the city, they encourage me, they say they are behind me, and I am speaking directly to them," Garcia told reporters on Saturday.
"Clearly, when you are ambitious and are champions of Italy, you want to defend that title tooth and nail."
Garcia stressed the need for his team to be more efficient in front of the goal.
"In Serie A we conceded five goals despite only allowing eight shots on target, so it’s not as if we are caught out of position," he added.
"The team must hold on to its principles, the high press, passing, possession, chances."
Garcia added that they will not underestimate Bologna as they head to Stadio Renato Dall'Ara.
"Bologna are a good team with a fine coach, I admired Thiago Motta as a player and am happy he is doing well as a coach too."