It is expected to be added to Public Health England's weekly watch list of "areas of concern", as officials try to combat a high infection rate.
Doing so would mean increased monitoring of cases and additional lockdown measures in future if infections cannot be brought under control.
Council leader and chair of the outbreak control board, Judith Blake, said: "This is a pivotal moment in our efforts to control the spread of the virus in Leeds and to keep our city open.
"Nobody wants to see further restrictions on life in Leeds and we want to assure everyone that we're doing absolutely everything within our power to avoid that happening.
"But the harsh reality is that if our infection rate continues to rise as it has been, we will be left with no alternative.
"With that in mind, now more than ever we need a collective effort from the people of Leeds who have shown so much resilience and civic pride throughout this crisis.
"It's up to us all to keep our families, friends and neighbours safe and to play our part in keeping Leeds's recovery going."
The city's infection rate has risen to 29.4 cases per 100,000 people, with 44 new cases identified on Wednesday.
The council has reminded people to isolate if anyone in their household is showing Covid-19 symptoms, to keep social distancing, wear a face covering and to wash hands regularly.
Victoria Eaton, Leeds City Council's director of public health, added: "It's imperative that we do all we can to contain the spread of this virus and protect one another at this crucial time for the city."
Other parts of West Yorkshire, along with East Lancashire and Greater Manchester, have been subject to tighter coronavirus restrictions for several weeks.
Prof Robert West, from University College London's Institute of Epidemiology, told the BBC: “The problem is that we didn't as a country as a whole use the opportunity that we had to get the infection rate right down to around one in a million which could have been done in the way they've managed to do it in Scotland for example.
"We are where we are, the problem is that this remains a very infectious virus and what that means is that if you're near someone, with someone who has it, there's a good chance you'll get it."
PHE also designates areas of enhanced support – which are provided with additional resources – and while areas of intervention, which will request help from the government to deal with rising infections.