Corey Platt-May, six, and two-year-old Casper were struck by a car as they crossed the road with their family in the Stoke area of the city on Thursday afternoon.
Both boys died after being taken to hospital. A damaged black Ford Focus was found a short time after the crash.
Robert Brown, 53, and Gwendoline Harrison, 41, have been charged under joint enterprise of two counts of causing death by dangerous driving, West Midlands Police said.
Brown, of Attwood Crescent, Wyken, is also charged with driving while disqualified, no insurance and otherwise than in accordance with a licence, while Harrison, of Triumph Close, Wyken, faces a common assault allegation.
They appeared at Coventry Magistrates’ Court, sitting at Leamington Justice Centre.
Corey and Casper’s mother Louise Platt-May paid tribute to the boys following their deaths.
She described Corey as “an amazing boy” who had a passion for football, while his younger brother Casper was a “cheeky little boy” whose “smile could light up any room”.
The boys’ aunt Chelsea Platt-May told Sky News they had been on the way to a family day out at the park at the time of the collision.
“I hope – it sounds horrible to say – they didn’t suffer and I hope they were going with happy thoughts and not thoughts of pain,” she said.
Both boys were taken to hospital with severe injuries. Casper could not be revived and died a short time later. Corey died despite undergoing surgery.
Family and friends have launched a number of fundraising pages to help their parents.
Corey’s football coach Matthew Hopkins told Sky News he was unsure how the youngsters’ teammates would react.
“I won’t be able to say ‘good tackle Corey’, ‘good pass Corey’ or ‘good goal Corey.’ That’s really sad,” he said.
Detective Sergeant Paul Hughes said: “It is always tragic to lose loved ones, especially when so young, and we are continuing to offer our support to family members at such a devastating time.
“We are very appreciative of the overwhelming help we have received from the public during our investigations to date and we continue to ask anyone with information who has not already spoken to us to get in touch with us.”