The Red Sox signed Lyle Overbay to a minor-league deal before spring training camps opened, in the hopes of having a useful glove and occasional bat off of the bench for first base and designated hitter. Overbay isn't the greatest hitter, nor the greatest fielder, but we're talking about a part-timer whose primary goal is to keep there from being a hole at either position on a day off for regulars.
Overbay's chances of making the club were very good, if only because there was no one else competing for that spot. That is, until the Red Sox brought in Mike Carp in a trade with the Mariners. Carp is on the 40-man roster, and Overbay is a non-roster invite to spring training. If Overbay makes the team, it'll be simple enough to fit him on the 40-man, as Carp could be designated since Boston only needs one of the two. Whether or not this is what will happen is up in the air, though, and likely won't be solved for another three weeks or so, when Overbay, as a major-league player signed to a minor-league deal, is told in advance whether or not he's made the Opening Day roster.
Let's say the Red Sox plan on keeping Carp, and Overbay isn't going to make the team. You can be optimistic and think he'll accept an assignment to Triple-A, but there isn't necessarily room for him there, either, since Mauro Gomez is around and on the 40-man. It's more likely the Red Sox ask him if he's okay with being traded to a specific team, to preemptively answer the question of whether or not he'll opt-out of his new club, and then send him off for cash. The Brewers, who were interested in Overbay before Boston signed him, are a fit now that their situation has changed and Mat Gamel is out. The Yankees, now sans Mark Teixeira for the next two months, are also a fit.
Obviously, the Red Sox don't want Overbay to go to the Yankees if they aren't using him. They thought enough of him to bring him to spring training, and therefore think he could be useful in the right setting. Overbay is a left-handed hitter, so you know he'd at least help out in Yankee Stadium, and with Teixeira out for the first month and change of the season, the injury-prone Travis Hafner at designated hitter, and, as of today, Dan Johnson the starting first baseman on the depth chart, you know they would be interested in picking him up if he's freely available. In short, Boston wants to make sure that either they (a) keep Overbay or (b) keep him away from the Yankees.
The first option shouldn't be forced -- the Sox should only keep Overbay for themselves if they think it's in their best interests. But the second has to happen if the first isn't the chosen path, or else there is a good chance the Yankees will be picking up the non-roster refuse of their rivals. Overbay would get at least a month as a starter to pick up consistent at-bats, would have a realistic chance of picking up playing time at DH... the promise for playing time that exists in Boston would be about equal to what he would see in New York. Anything that keeps the Yankees from reeling just a little longer isn't good for whatever chances the Red Sox might have in 2013, and losing Overbay to pinstripes qualifies.
The Boston front office is surely aware of all of this, though, and aren't likely to give Overbay the chance to pick his next destination as a free agent. He'll either be on the Red Sox or sent packing before he has the option to be otherwise, and hopefully the Yankees are still stuck looking at Dan Johnson at that point.