Relatives of Kailash Chander, ex-mayor of Leamington Spa who is assisting police with inquiries, said the crash was "beyond anyone's imagination".
Passenger Rowan Fitzgerald, seven, died when the bus crashed into Sainsbury's on Trinity Street on Saturday.
Pedestrian Dora Hancox, in her 70s and from Nuneaton, was also killed.
Mr Chander has not been arrested in connection with the crash.
When the crash happened Rowan, of Leamington Spa, was on the upper deck with his grandparents and his cousin, an eight-year-old girl, who was seriously hurt. She remains in a stable condition at Birmingham Children's Hospital.
Five other people required hospital treatment.
A statement issued by unnamed relatives of Mr Chander said: "Being a grandfather himself, he is devastated to the core by what has happened.
"He is a very quiet man, and a caring and devoted person. He has strong ties with the local community and has spent most of his life helping and thinking of others."
The relatives said they would "only ask that the privacy of the families is respected at this tragic time" and the police "are given space to complete their investigation".
"We want to express our deepest and most sincere condolences to those involved, particularly the families who have lost their loved ones," it concluded.
A friend of the driver, Baldev Singh Sandhar, said: "He is such a good character. He always helps people. He never likes to see any people suffering from anything."
A former colleague of Mr Chander, who was mayor in 2008, said he had been left "deeply traumatised".
Rowan was a pupil at St Anthony's Catholic Primary School where pupils are being supported by staff.
Head teacher Jane McSharry said he was "the most beautiful little soul".
She said: "Rowan was a perfect, kind, unassuming, gorgeous little boy, with a smile to melt any heart.
"He loved to learn, but he also loved to play. So he loved his sport, he loved his football."
Insp Paul Bennett said as the families of Rowan and Ms Hancox "struggle to come to terms with their loss, they have chosen at this time not to release any images of their loved ones or to provide any words of tribute".
"They have also asked for the media to respect their privacy as they grieve."
The bus company, Stagecoach Midlands, said: "No driver we employ is allowed behind the wheel of our buses without us being satisfied that they are fully qualified, safe and meet all necessary legal requirements.  In this case, our driver has many years' experience and has also passed a mandatory annual medical.
"Both our driver and our company are co-operating fully in helping police determine the full circumstances involved. Police have not yet completed their investigation and it is inappropriate to speculate in advance of their inquiries being concluded."
Almas Fatima from Pleck, Walsall, said her two daughters, aged 12 and 14, were not able to sleep after they saw the crash and narrowly missed being hit.
"I just remember a bus coming towards us and we survived. That's the only thing we remember," she said.
"We were walking on the pavement and as soon as the bus started moving it hit the other bus as well as we realised it was not stopping, it was coming at full-speed towards us, and I told my daughters, 'just run'."
The bus hit a number of cars and a lamppost before crashing into the open supermarket at about 18:00 BST on Saturday, officers said.
The force has appealed for witnesses to come forward.
Insp Bennett said: "Investigations are ongoing but we are aware of a number of vital witnesses who we are yet to speak to.
"We know there were possibly hundreds of people at the crash scene on Saturday evening and I would urge these people to come forward as soon as possible and give statements. Also, if anyone has images or footage of the scenes, please contact us.
One shop owner said roadworks had caused more confusion at the "congested" junction.
Rupak Parvakhar, who owns the City Chippy on Fairfax Street, said temporary traffic lights had confused some pedestrians.
"Most of the time the poor pedestrians are crossing and whether the lights are red or green they are panicked and confused as to what they are doing," she said.
"Me, on so many occasions, I'm trying to weave my way through the people and then you have buses coming down Corporation Street on a really tight bend, trying to turn."
The emergency services worked into the early hours on Monday to remove the bus and secure the area.
The family of a 77-year-old bus driver whose vehicle crashed into a Coventry supermarket, killing two people, say he is "devastated to the core".