The 10-year-old youngster’s family contacted local MP Jo Swinson in alarm after their daughter told them she did not want to go outside .
The vile phrase was coined in anonymous letters circulated last month that suggested people could win “points” for violent acts against Muslims.
Ms Swinson said: “It’s deeply disheartening that a 10 year old girl is afraid to go outside, just because of her faith.
“The impact of rising bigotry in our country is heart-breaking. It’s important that we all send love to Muslim friends and fellow citizens.
“The idea of a ‘Punish a Muslim Day’ is abhorrent.
“Yet in spite of this communities have shown huge strength in their response to such hatred and in their support for one another.
“Now the Scottish and UK Governments must ensure that the people behind such appalling behaviour are tracked down and feel the full force of the law.”
A police investigation has been taking place since the letters and online images circulated.
Communities came together on Tuesday to condemn the campaign and express support online for Muslims with counter-hashtag #LoveAMuslimDay trending.
Last month, hundreds of people marched in Glasgow as part of a Stand Up to Racism demonstration.