Nonis has been in negotiations with center Tyler Bozak, who last pitched an eight-year deal worth just under $5 million per season, but it's believed the Leafs are balking at the asking price.
Instead, the Maple Leafs are targeting center Stephen Weiss and right wing David Clarkson, both of whom are unrestricted free agents.
Weiss, 30, has spent his entire career with the Florida Panthers, but is looking to move on, and the Leafs are one of the teams interested in him.
He's believed to be seeking a five-year deal worth around $25 million.
Clarkson, 29, has spent his entire career with the New Jersey Devils and recently rejected an offer pitched by the team last week.
Various reports suggest Clarkson could command upwards of $6 million per season, which might be too much for the Leafs' liking.
Meanwhile, the Leafs are also shopping center Mikhail Grabovski and defenseman John-Michael Liles.