The Memphis-based pulp and paper company has been pursuing the Irish packaging company for over three months.
International Paper has until 7:00 AM local time on Wednesday to make the final call, either offering the firm a deal or dropping the bid entirely.
If the company decides to abandon the bid, Irish law dictates that it cannot try to acquire the company again for another 12 months.
The U.S. company had made two takeover proposals in February and March but were rejected both times.
Shares of International Paper were down less than 1% in Tuesday after-hours trading while Smurfit Kappa shares in Dublin fell nearly 7%.