More than 20 local museum leaders have joined Mr Butler in signing the open letter.
“None of us think there is a magic wand or fairy godmother to sort out regional museums,” he said.
“The best thing you can do is to come and visit us, enjoy our museums or even lobby the government.”
In response to the letter, a Department for Digital, Culture, Media and Sport spokesperson said: “We recognise the pressures they face, and we are committed to giving stability back to local councils so services such as these can best meet the needs of their communities, alongside continued support provided to regional museums via Arts Council England.”