Logan accepted a $3.15 million, non-guaranteed deal in advance of Friday's arbitration deadline, leaving Robertson as the club's only remaining potential case.
Robertson filed for $3.55 million, while the Yankees filed at $2.85 million.
The 27-year-old Robertson earned $1.6 million last season, going 2-7 with a 2.67 ERA in 65 relief appearances. The right-hander is arbitration eligible for the first time.
The Yankees hammered out agreements with pitchers Phil Hughes ($7.15 million) and Joba Chamberlain ($1.875 million) this week before settling with Logan.
The left-handed Logan led the American League with 80 appearances and was 7-2 with a 3.74 ERA and one save last year, earning $1.875 million in 2012.
The relatively small gap between the Yankees' offer and Robertson's request suggests that a deal could be reached before the scheduled period of Feb. 4-20, when remaining cases will be settled before arbitrators.
The Yankees have not had a case reach the arbitration stage since 2008, when they defeated pitcher Chien-Ming Wang in a hearing over a difference of $600,000.