Blackpool chairman Karl Oyston has apologised "unreservedly for any offence" caused by abusive text messages he sent to one of the club's fans.
Oyston says he became embroiled in a furious row with supporter Stephen Smith, reportedly calling him a "retard" and adding that he should "enjoy the rest of your special needs day out."
The club's supporters' group are calling on him to resign.
"I regret stooping to the level of those threatening and abusing my family," said Oyston in a statement.
Blackpool are bottom of the Championship, a full 10 points from safety after a torrid summer and first half of the season.