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I don't know the exact specifics of what triggers this (or even if there are any), but there have definitely been times where I have received messages that do state what another team has offered as a way to encourage me to make a similar or better offer. As I recall, they tended to be situations in which the player definitely wanted to resign with my team (i.e. I received a message requesting an extension before they hit free agency). Other people may have seen this in other situations though.
One thing you might do though is to look at what the player is asking for. Often times this will change over free agency, presumably in connection to offers received from other teams. This is different than the general trend of declining demands as free agency moves into the preseason and spring training; the demands actually go up in this case. In general though, you rarely need to add more than a few hundred thousand (in the modern era) per year to become the preferred offer again.
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