There's more than enough room for both pitchers to stay.
According to Yankees closer David Robertson, the team has yet to reach out to him about a potential extension. For what it's worth, if you read between the lines, it sounds like he'd be open to an extension, though he's probably not going to give them any discount by saying "there's just been zero talks. When the offseason comes, it comes and we'll hear what other teams and everybody else wants to say." That sounds more like he's ready to test free agency and see who gives him the best offer.
The 29-year-old strikeout machine will be a free agent at the end of the season, and though they have Dellin Betances to take his place, if that is actually the direction they want to go in, the Yankees are better off with both in the bullpen going forward. The way this usually works is the veteran closer goes out and gets himself an expensive contract and the new young closer comes in and makes the league minimum. It makes perfect sense in terms of strategy, just not in this case. They could let D-Rob walk, save the money for something else, and use Dellin, but they don't have anyone to really replace Dellin and they don't really need to save money. These are the Yankees, they can have their cake and eat it too. Just keep both assets and have a better bullpen.
Think about it, who exactly is in the Yankees bullpen that has been that impressive this year? It's just Dellin and D-Rob. Adam Warren has a very pedestrian 4.58 ERA since June 1 and batters are hitting .311/.370/.432 against him over his last 18 innings. That doesn't sound like a bullpen building block for next year. Shawn Kelley has dealt with injury and ineffectiveness, you have Matt Thornton, who has been hit-or-miss this year, and then a rotating supply of randoms. They will all be here next year, but take away Robertson and you're left with Dellin-Kelley-Warren? That's not very promising.
Jose Ramirez, Preston Claiborne, Chase Whitley, and Shane Greene would likely all be in the running for a spot in next year's bullpen, but how effective would that corps or relievers be? Sure, you have Jacob Lindgren and Tyler Webb coming through the pipeline, but Mark Montgomery should be a cautionary tale for putting too much faith into a relief prospect. Betances can be an effective closer, but that's really all the faith I would have in that entire bullpen. They can afford to re-sign Robertson and take on the risk of a reliever on a long-term contract (not too long) and they need him to push the pecking order down one rung and keep some very questionable arms, like David Huff or Jim Miller, out of the bullpen.
They locked up Brett Gardner this past offseason, proving that they were open to the practice, however, their disinterest in doing the same with Robertson doesn't seem to be too thought out. Yes, there is a certain amount of risk with a reliever, but they really don't have much of a choice at the moment. You could say that the Yankees can easily add a veteran reliever on a free agent contract, but they won't. The Yankees don't spend money on external relievers unless they throw with their left hand or are mandated by the Steinbabies. Even with Betances, the Yankees need Robertson and that's not really a bad thing.
Now we're midway through the season and the team seems to be hovering in mediocrity; it might be nearly time to start thinking about next year. They broke down and extended Gardner mid-contract, but could the next step forward be a midseason extension?