Born in Moncada, Valencian Community, Iborra was a product of Levante UD's youth system. He made his first-team debuts during the 2007–08 season, as the club's severe financial problems prompted the exit of several players. After appearing in a Copa del Rey match against Getafe CF on 9 January 2008, he played in La Liga against Real Madrid four days later (0–2 home loss).
On 30 March 2008, Iborra scored his first Levante goal, a last-minute effort in a 1–2 loss at UD Almería. In the 2009–10 campaign, as the side returned to the top flight after a two-year absence, he was one of the team's most used players — 36 games, 2,640 minutes — netting once.