Blackburn had 47 cases per 100,000 people in the seven days to 11 July, up from 31.6 cases per 100,000 in the seven days to 4 July, according to figures published on Tuesday.
The authority is third on the list of highest weekly rates, behind Leicester, which has a rate of 118.2 cases per 100,000 and is subject to a local lockdown, and nearby Pendle, with a rate of 76.6.
If rates continue to rise in the next few weeks, the council would consider reimposing some lockdown restrictions, Mr Harrison said.
Residents are being told to wear cloth face coverings in all enclosed public spaces, including workplaces, libraries, museums, health centres and hair and beauty salons.
Blackburn with Darwen has introduced extra lockdown measures including a limit on the number of people allowed to visit a household after a rise in Covid-19 cases.
The Lancashire authority’s director of public health Dominic Harrison announced the new measures on Tuesday, to be followed for the next month with the aim of avoiding a local lockdown.
The restrictions include a limit of two people from the same household allowed to visit another home.
Mr Harrison also called for people only to bump elbows with anyone outside of their immediate family, instead of shaking hands or hugging, and said public protection advice for small shops was being stepped up to ensure measures including social distancing and increased ventilation were being followed.
Targeted testing is taking place in the borough, with a mobile testing unit at Witton Park Academy as well as a facility at Royal Blackburn Hospital, and residents have been told they do not need to have symptoms to be tested.
“These steps will help and we are appealing to everyone in Blackburn with Darwen to follow them to protect themselves and their loved ones,” Mr Harrison said.
“If we don’t, a local lockdown, like in Leicester, becomes a very real possibility.
“The council is working with the NHS and the wider community on this, it’s up to us all to help each other stick to the advice that’s been given.
“We can only avoid further lockdown measures if we work together.
“In the next week to 10 days, we will see a rise in the number of cases in the borough due to the increased targeted testing we are doing in communities most at risk, and because we have opened a Pillar 2 testing site next to the hospital last week.
“In another two weeks, if the rates are continuing to rise, we will have to consider reversing some of the national lockdown lifting measures locally one by one until we see a reversal in the current rising trend.
“It’s up to everyone to make sure we don’t have to do that.”