The offer is worth $15.3 million over one season. Robertson now has one week to accept or decline the offer.
Robertson had a 3.08 ERA, 39 saves and 96 strikeouts in 64.1 innings in 2014. The 29-year-old is a former 17th-round draft pick by New York in 2006.
Kuroda, 39, was 11-9 with a 3.71 ERA for the Yankees in 2014. In three seasons in New York, he was 38-33 with a 3.44 ERA and 463 strikeouts.
On Monday, it was reported that the Yankees are in talks to re-sign third baseman Chase Headley, who was acquired in a July trade.
The Yankees went 84-79 last season, missing the playoffs for the second straight year.
- Molly Geary