NSPCC Scotland wants key worker status for their counsellors after 167 children north of the border called the helpline in one week.
Counsellors in the Glasgow office spoke to 80 children anxious about the outbreak between 16 and 22 March, with the Aberdeen base counselling 87.
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Among the concerns raised were struggles accessing mental health support, problems caused by isolation and arguments at home.
One teenage girl told a counsellor: "I feel really anxious, upset and lonely. The news has made my mental health worse but my Camhs appointment has been cancelled and school has closed.
"I'm stuck at home having a horrible time because my sisters are bullying me because I'm autistic."
Across the UK, 913 children and young people called the helpline over coronavirus fears, with calls hitting a peak of 121 on 18 March, the day Nicola Sturgeon and Boris Johnson confirmed UK schools would close.
Girls aged 12-15 were the most likely to call with concerns about the Covid-19 pandemic.
NSPCC Scotland is in talks with Scottish government and local authorities to give Childline staff key worker status to help support children through the outbreak.
Leanne Ferries, Aberdeen Childline service manager, said: "Our staff and volunteers have been working tirelessly to keep Childline running, and in this public health emergency it's never been more important for children to have somewhere to go to get advice and support.
"Coronavirus is in the news constantly, causing some young people to be anxious, particularly those who are already coping with other issues in their lives. Things are being made even more difficult for them by schools closing, being confined to their homes and not being able to see their friends.
"That is why it is vital that a service like Childline is there throughout this crisis, always ready to listen and help.
"Keeping children safe and providing them with a space to talk about their concerns is our number one priority."