The Cobra meeting will be attended by ministers from the Department for Environment, Food and Rural Affairs, Home Office, and Department for Health, as well as the Environment Agency.
A No 10 spokesperson said the government understood firefighters, farmers and NHS staff were all facing challenges connected to the weather.
There has already been one Cobra meeting on heatwaves this year chaired by officials, but this will be the first such emergency session chaired by Burnham.
In June, the government under previous prime minister Sir Keir Starmer announced that specialist firefighters would be positioned in key locations across England in response to wildfire risks, with forces encouraged to use a government fund to upgrade equipment.
Burnham had faced calls from the Green Party to chair a Cobra meeting on the heatwaves, with leader Zack Polanksi accusing the prime minister of inaction amid threats to public health and food supplies.
Speaking on Wednesday, Polanski welcomed the meeting but said it was “not enough just to discuss” the threats from heat.
He added that recent temperatures showed the need to offer workers flexible hours to allow them to avoid the hottest part of the day, and for a legally-enforced maximum temperature for workplaces.
The party has previously called for more publicly accessible “cool spaces” for vulnerable groups and financial support to help households heatproof their homes.
Fire chiefs have said that wildfires are becoming a “recurring and growing feature of the risks facing our communities” amid a changing climate.
Crews responded to 393 wildfires across England and Wales in July, making it the busiest month on record.