Sturridge posted a video to Instagram displaying the damage done, along with CCTV footage of the three hooded men responsible, before offering his followers a large reward for anyone who can help aid the Pomeranian's return.
The England international, who has since posted four images of Lucci, said the crime was committed during the two hours he went out.
"Whoever brings my dog back – 20 grand, 30 grand, whatever," Sturridge said.
"You ain’t going to get more than that anyway, and if you are, let me know. I’ll pay more than them.
"Whoever knows who broke into my crib, I’ll pay you anything, I’m dead serious.
"It doesn’t matter the cost."
A UEFA Champions League winner with Liverpool and current free agent, the 29-year-old has since been linked with a move to newly promoted Premier League side Aston Villa.