Well dang. That went far better than I expected. I had three players in that bucket, all Type B eligible, I offered their asking prices plus an extra season and they all took the first offer. Might be I'm just lucky this time. For reference, their Patience ratings are 58, 65, and 99.
I suspect if they were Type As the result might be more like what longballwins suggested, but that's just conjecture at this point.