Shares of Bemis, which has a market value of about $4.20 billion, closed up 11.27 percent at $51.53 on Friday.
A deal could be announced as soon as next week, according to the report.
When asked by Reuters about the possible acquisition on Saturday, a spokesman for Amcor declined to comment.
Bemis was not immediately available for comment.
Reporting by Sanjana Shivdas in Bengaluru; Additional reporting by Will Ziebell in Melbourne; Editing by Nick Macfie