Talks have taken place this week.
Oxfordshire's Director for Public Health and Council Leaders have revealed that they're pushing for the whole of the county to be moved to a high alert level 'as soon as possible.'
The officials say that the virus has spread to a wider age range and hospital admissions have begun to increase as a result.
Oxfordshire is currently at the 'medium' or tier 1 level in the COVID-19 alert system - which classifies areas as medium, high or very high based on their numbers of infection and overall risk level.
Moving to the 'high' COVID-19 alert level would mean that residents can't socialise with anybody outside their household or support bubble in any indoor setting, whether at home or in a public place.
Oxfordshire County Council's Director for Public Health Ansaf Azhar said: "Across all areas of the county, we are starting to see a significant shift in the spread of the virus beyond people in their teens and 20s to older and more vulnerable age groups. This is a really concerning development. We know that, once the virus starts to spread to more vulnerable groups, then hospital cases will rise and deaths will inevitably follow.
"We have seen what's been happening across the north of England and how the virus has quickly taken hold across huge swathes of the community. Based on the current trajectory of the virus, we could well find ourselves in a similar position in just a few weeks' time if we do not take collective action now."
"With half-term approaching, as well as events such as Halloween, Bonfire Night and Diwali coming up, it's very easy to get caught up in the excitement of meeting up and celebrating with friends and family. But we mustn't forget about COVID. We need to do everything we can to keep our families and communities safe and stop the spread.
"I know the temptation will be to meet up and socialise over half-term. However, the virus thrives when people are in close contact with one another. So I would strongly urge everyone to limit their social interactions and focus instead on the many COVID-secure family activities that are taking place over half-term."
Moving to a high alert level would mean that residents could not socialise with anybody outside their household or support bubble in any indoor setting, whether at home or in a public place.
Leader of Oxfordshire County Council Ian Hudspeth said: "In light of the escalating situation across the county, we are pushing hard for Oxfordshire to be moved to a high alert level. This would be a preventative measure to stem the spread of the virus and protect the county's most vulnerable residents.
"We are aware that some businesses, particularly the hospitality sector, would be affected if we moved to a high alert level, and we welcome the announcement from Government today that additional support will be available.
"However, we must do everything we can to keep residents across the county safe. We know that the majority of transmissions occur when different households mix, so increasing our alert level to high, which prevents households from mixing socially in indoor settings, is one of the best ways we can help our residents at this critical time."