The Penguins have reportedly offered the Canucks a 2014 first-round draft pick, a 2014 third-round draft pick, center Brandon Sutter and Vancouver's choice of defensemen Simon Despres or Brian Dumoulin.
The Tribune-Review suggests another player from both teams could be involved, though Shero's primarily targeting Kesler.
Shero indicated in February that he is not interested in giving up assets for a rental player, but Kesler is locked up for two more years, beyond this season.
The paper claims Shero first inquired with about Kesler in January.
If the Penguins are unable to acquire Kesler by the March 5 trade deadline, and the Canucks don't trade him to another team, Shero will try to complete a move in the off-season.
The Detroit Red Wings, Chicago Blackhawks, Columbus Blue Jackets and Philadelphia Flyers are also interested in Kesler.