Renner, 44, is concerned that Pacheco, who filed for divorce on December 5 and demanded physical custody, will take off to Canada with his beloved daughter, reported Us magazine.
In the 23-year-old actor's divorce filing, Pacheco asked that Renner should return her "stolen passport".
The couple's marriage started to crumble soon after their secret January 2014 nuptials.
"The two had started sleeping in separate rooms some time ago. It wasn't traditional, but he was hopeful they could make it work for Ava. He's devastated," a source said.