It has been done in MLB.
In 1967, Gil Hodges was the manager of the Washington Senators. Shortly after the season ended the Senators traded him to the New York Mets for Bill Denehy and $100,000 cash. The Senators were strapped for cash and Hodges probably wasn't going to stay with Washington more than one more season. As with all trades, MLB must approve it and they had no problem doing so.
Lou Piniella managed the Seattle Mariners in 2002 and left to manage the Tampa Bay Devil Rays in 2003. Shortly after Piniella joined Tampa they traded Randy Wynn, one of their best players, to the Mariners for a minor league prospect. The trade was believed to be compensation to the Mariners for Piniella leaving to join the Devil Rays.
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yes they can...back in 2002 the Oakland Raiders traded Jon Gruden to Tampa Bay for two 1st round draft picks ('02,'03), two 2nd round draft picks('02,'04) and cash ($8 million).