"Phil Jackson won't go to Brooklyn," the tweet said. "He'll only go somewhere he can win a title. Not interested in good players, he wants great players."
Jackson, the former Chicago and Lakers coach, has been rumored as a possible successor to Johnson.
The tweet was quickly removed from the feed, which is generally reserved for team updates. A spokesman said the Jazz's marketing department operates the Twitter account, but declined to say how many people have posting privileges.
In the statement the team said, "The Jazz does not comment on other NBA teams transactions and furthermore, does not condone any negative comments about players, coaches, or front office staff throughout the league. It is simply unprofessional and unacceptable."
A Nets blog, Nets Take Over, posted a screen shot of the tweet, and wondered if the comment was a shot at former Jazz point guard Deron Williams, now in Brooklyn. Williams publicly panned Johnson's offense last week when the Jazz were in Brooklyn, saying he was more comfortable in the Jazz system.
Swell news for Williams
For Mo Williams and The Great Thumb Affair of 2012, the focus has turned to swelling, which Williams said has mostly subsided. Sidelined for the third straight game Friday with a severely sprained thumb, Williams now looks toward a possible re-evaluation when the team returns from Los Angeles next week.
"Still got some pain in it," he said, "a lot of pain. I wouldn't be able to catch a ball, stuff like that yet."
Williams aggravated the thumb, which he sprained earlier in the season, on Dec. 22 in Miami as he defended a play where Mario Chalmers drove the lane.
The last week the Jazz co-captain has been reduced to spectator.
"The most frustrating thing is you see your teammates out there playing and fighting," Williams said. "And good or bad you can't go out there and help."