They say the payments will allow disadvantaged parents to go out and buy groceries "without stigma" during the Covid-19 crisis.
The calls come in a letter being sent to Scotland's First Minister later today, after it was announced schools across Scotland will close by the end of this week.
Closures could continue until the summer holidays, potentially not re-opening before the start of the autumn term in mid-August.
Around 240 000 children are already officially recognised as living in poverty in Scotland.
There were nearly 270,00 pupils registered to receive free school meals last year, with a take-up of more than 188,000.
Free school meals can save families £20 a week for each child, and can often mean the difference between eating and not.
Child Poverty Action Group in Scotland is leading the calls, with support from organisations including Save the Children, Children in Scotland, Action for Children, Nourish Scotland, One Parent Families Scotland and the EIS teaching union.
John Dickie, director of Child Poverty Action Group in Scotland, says many children could be forced to go without food as vulnerable households suffer the impacts of emergency health protection measures.
He said: "It is absolutely vital that government at every level acts to protect children from the impact of Covid-19.
“Here in Scotland we welcome the commitments that were made in Holyrood today but we now urge the First Minister to ensure low-income families receive cash payments to help meet the extra costs of having children at home every day.
“Hard-up families are already struggling to stay afloat. This payment would provide a lifeline in these exceptional circumstances."
The letter states: “The additional cost of providing lunch will be a significant, and in many cases unmanageable, financial pressure on families.
“Low-income families are already struggling to make ends meet and free school meals play a vital role in relieving pressure on family budgets.
“Building on the welcome support announced today by the cabinet secretary for communities, we believe the most effective way to protect families from the additional pressure that loss of free school meals will create will be to make cash payments to families currently in receipt of free school meals.
“We believe support in the face of the Covid-19 crisis must be provided in a way that gives families the choice and agency to meet their family's needs in this exceptional time. A cash payment in lieu of free school meals would be a dignified response, respectful of human rights and avoiding any potential stigma.”
Nicola Sturgeon announced today that schools and nurseries across Scotland would likely close to pupils from the end of this week.
She said the Scottish Government would continue to provide food to children who receive school meals.
“This has been one of the hardest decisions we have faced so far as we tackle the coronavirus,” she said.
“There will require to be a lot of local flexibility and we are working closely with local authorities to put those arrangements in place.
“Teachers, school staff and those in the nursery sector will have a vital role in the weeks ahead and we will work with you to minimise the impact on all our young people’s education, and in particular the most vulnerable groups.”
A £70 million ‘food fund’ has been announced among a £350 million emergency package aimed at supporting welfare and well-being among those worst hit by the pandemic.
The Scottish Government has said it is working closely with councils to mitigate the impact on three groups as a priority: vulnerable pupils and those receiving free school meals; pupils undertaking coursework and preparing for exams; key workers including doctors, nurses and emergency service workers who have children.
Education secretary John Swinney is due to set out measures being put in place to MSPs at the Scottish Parliament tomorrow.