One of the best left handed relievers on the market is former Blue Jay Scott Downs, but according to Ken Davidoff of Newday the Yankees have no interest in him because he is a type-A free agent. That means they would have to give up a first round draft pick to sign him which they think is just too much to pay for a middle reliever.
It’s likely that the Yankees will actually have to give up their first round draft pick if they sign Lee. So if they did sign both Lee and Downs it would mean that they would have to surrender their second round pick to the Blue Jays. That might sound better, but even in that scenario they are unlikely to want to give up the pick.
The Yankees know that most of the best talent in the drafts are in the early first couple of rounds. Without their first two picks it would make having a successful draft very difficult. They wouldn’t want to put themselves in that position for a middle reliever.
As good as Downs has been in recent years, middle relievers are just to volatile. Even the best ones have huge up and down years. So giving up a draft pick for one is too much of a gamble unless your bullpen is in dire need.