Mexican-born Tavano is not classed as an import and spent two seasons in the youth system at Liga MX club Santos Laguna.
His move follows that of Emiliano Tade, who returned to Argentina to play for home-town team Club Atletico de Mitre.
Defender John Irving has trialed with League Two club Portsmouth and will return for another in July once the English season has ended.
"It's going to make it very hard but it's the way it is," Auckland City coach Ramon Tribulietx said. "On the one hand, we are delighted for these players that they get to fulfil their professional dreams. On the other hand, it's very difficult to lose key players at this stage of the season, especially with our structures and the way we play. It's not that easy for someone else to slot in."
Of everyone at Auckland City, Tribulietx was probably the subject of most speculation, as architect of the Club World Cup success. He admits to attracting plenty of overseas interest but he will not consider anything until the end of the season.
"I've got a job to do here at Auckland City. We have the league to focus on and then, hopefully, the Oceania Champions League, which will be extremely tough this year, being played in Fiji. And as I say, I love my job at Auckland City, so it would take something pretty special to make me consider leaving."