Rossi claims the Penguins have offered center Brandon Sutter, a pick of young defensemen Simon Despres or Brian Dumoulin, a 2014 first-round draft pick and a 2014 third-round pick in exchange for Kesler.
Shero is believed to be willing to pay a higher price, if need be.
In an interview on The Team 1040 radio station in Vancouver, Rossi said, "Ray Shero has told Vancouver he'll best any offer they get (for Kesler)."
Multiple sources have confirmed to TFP that Kesler would waive his no-trade clause to go to the Penguins.
Several teams have expressed an interest in Kesler, including the Detroit Red Wings, Chicago Blackhawks, Philadelphia Flyers, Toronto Maple Leafs, Columbus Blue Jackets and Los Angeles Kings, among others.
TFP has learned that some teams, including the Maple Leafs and Blue Jackets, have been told Kesler will not accept a deal to their team.
The Canucks are not forced to trade Kesler by Wednesday's 3pm ET trade deadline, and could resume trade discussions in the off-season, but a deal is looking more and more likely.
Vancouver is in Phoenix on Tuesday, their last game before the deadline, and possibly Kesler's last in a Canucks uniform.