Councillor Anna Richardson, city convener for sustainability and carbon reduction, said: "We know that these are hard times for anyone delivering front line services".
Deputy leader of Hampshire County Council Rob Humby said: "Last weekend over 29,000 people visited Hampshire HWRCs - to put that into context, our busiest Easter bank holiday weekends might see a maximum of 22,000 people".
Mel Amara-Carnell, 46, posted the idea in her street's WhatApp group.
RMBC said it would empty as many bins as possible but may not be able to cover the whole borough.
"These circumstances have brought us all together again".
West Sussex's rubbish tips will be "closed until further notice" due to the coronavirus pandemic, it has been confirmed.
Afton March household waste recyling centre to the public.
Much of the waste that goes through the recycling centres is bulky waste (large items such as furniture, wood and metals) and garden waste.
Penny Carpenter, chairman of the environment committee, said: "During this challenging time for everyone, it is important we resource our most important services".
A number of councils have announced changes to their recycling collections as staff become ill or are forced to self-isolate, while many more are warning residents there could be disruption to services.
The council is reminding people that it has limited capacity in its collection vehicles and at waste transfer stations, and so it needs Dorset residents to help. Please do not squash waste down into your bin after you've put it in there as this can prevent it falling out during the emptying process.
Manchester City Council said food and garden bins would be collected fortnightly instead of every week and should be put out alongside general waste.
Community recycling centres are also closed.
The council will continue to work closely with Glasgow City Health and Social Care Partnership, NHS Greater Glasgow and Clyde and Health Protection Scotland to monitor and respond to the national situation.