Cani, an Albania international who plays for Catania in Italy's Serie B, is likely to join the Whites until the end of the season, according to sources from Sky.
United had also been targeting Leonardo Pavoletti, but the Livorno striker signed for Genoa on Friday.
The Albanian frontman joins Souleymane Doukara and defender Giuseppe Bellusci in making the switch from the Italian second tier to the Championship this season.
Leeds have only been able to sign free transfers on wages less than £600,000 per year after the Football League found the club in breach of Financial Fair Play regulations.
"I heard about a month ago that Leeds wanted me," Cani told leedsunited.com. "I was very happy about that and I’m very happy to be here."
Owner Massimo Cellino ruled out re-signing Norwich’s Luciano Becchio, who scored 86 goals for Leeds before leaving for the Canaries two years ago.