The former director has gone public with questions on how the club is being run, including whether ex-chief executive Charles Green is still an influence.
Mr Somers has sent a letter to Mr King criticising his "very damaging" statements and "innuendos".
The chairman has yet to respond to questions raised by Mr King and the Union of Fans - a collaboration of supporters groups - including an explanation of what has happened to Mr Green's holding in Rangers International Football Club (RIFC).
Mr Somers told Mr King: "These statements and innuendos are very damaging to the club, which we can only assume is your intention. We wish you to attend a meeting to explain your allegations in more detail as soon as possible."
The letter was released after the club issued a denial of rumours that RIFC was about to enter administration.