The sides "had extremely fruitful talks" during the three-day negotiating period and could soon reach a deal, Garafolo reported.
Fasano, 28, had a career-high 41 catches for Miami last season and caught five touchdown passes each of the past two years. He joins tight end Tony Moeaki to give new Kansas City quarterback Alex Smith two solid targets over the middle.
Kansas City released tight end Kevin Boss last month.
After two seasons with the Cowboys, Fasano spent the past five in Miami, catching a career-high seven touchdowns in 2008.
For more NFL news, rumors and analysis, follow @EyeOnNFL on Twitter, subscribe to our RSS Feed, watch Pro Football 360 daily at 3 ET and NFL newsletter. You can also follow Larry Hartstein @LarryHartstein.