South of Scotland council leaders say the area should be placed into Level one when Covid restrictions are reviewed on Tuesday.
Dumfries & Galloway and the Scottish Borders are currently in Level two, but with infection rates falling the First Minister could announce an easing of restrictions on the hospitality sector.
During last week's coronavirus update in Holyrood, Ms Sturgeon said the Scottish Government will move the area into the second lowest alert level when it believes it is safe to do so.
Nicola Sturgeon said: "When you have an area like Dumfries and Galloway bordered by other areas with not just slightly higher prevalence but in some respects quite a bit higher prevalence, that's just another reason for caution."
Speaking head of Tuesday, leader of Scottish Border's council Shona Haslam said: "Our numbers are low, they've always been low. The only indicator that was keeping us in level two was our test positivity rate - that has now fallen into level one.
"We are very clear with the Scottish government - and I was clear to the cabinet secretary yesterday - that everyone in the Scottish Borders believes we should move down to level one."
Councillor Elaine Murray, leader of Dumfries and Galloway Council, told ITV Border: "We are very hopeful that we'll enter level one tomorrow in the announcement that the First Minister will make to parliament.
"We have been within the criteria for level one since 11 November - almost a month. Therefore, we think we have proved that we are sufficiently in control of the virus here in Dumfries and Galloway."
The First Minister will make her announcement on Tuesday afternoon.