Hibs saw their 15-year stint in the top flight ended courtesy of a penalty shootout defeat to Hamilton Academicals in the playoff final second leg at Easter Road today.
The result means that Butcher has won just six of his 28 games at the helm but, contrary to internet rumours before today's match, the former England international insists that he wants to remain at the club.
"If it's down to me I will be here. There was stuff on social media that I would be quitting no matter what the result was. But that was never the case. I don't know where that came from," he told reporters.
"I want to continue, because I would like the opportunity to be able to restructure the club in the right way. I think that situation is out of my hands now."
Butcher was the subject of the fans' ire after the match as around 500 supporters protested against him and the board.