The colourful and easily-recognisable co-owner of the popular Pellegrini's Espresso Bar loved Melbourne and its people, and they loved him.

Mr Malaspina's family has accepted the offer of a state funeral for the much-loved 74-year-old, who was out for his daily walk when he was stabbed to death in Friday's terror attack.

"Dad was a larger-than-life character and his natural generosity made every individual he met feel welcome and important," the family said in a statement released by police on Monday.

"He was a distinctly recognisable figure in his colourful choice of clothing and his flamboyant personality.

"To us he was just a beautiful man, a loving husband and a doting father."

For Mr Malaspina, it was a joy to go to work every day.

"Each customer of Dad's was given the same respect and service regardless of their standing in society," the family said.

"To him everyone was special. His love and passion for his customers remained the same for over 40 years."

The family has been deeply moved by floral tributes amassing on the Pellegrini's doorstep and other gifts and condolence messages.

"To the people of Melbourne, your heartfelt tributes have shown us why our dad loved this great city so much."

Mr Malaspina was killed and two other men were injured when when 30-year-old Hassan Khalif Shire Ali drove a burning ute into the city and launched a knife attack.

The restaurateur died in the street while police shot Shire Ali who died later in hospital.

Mr Malaspina's family paid tribute to the bravery and heroism of the police and other emergency workers who put themselves in harm's way to protect others.

Premier Daniel Andrews said the state funeral for the Victorian icon will be held next week to remember Mr Malaspina's life on Bourke Street, not his death.

"Known for his welcoming smile and customary cravat, the 74-year-old holds a place in the hearts of all who knew him," Mr Andrews said.

Ideas are being considered for a lasting tribute to Mr Malaspina, including renaming the laneway next to Pellegrini's.

"I know there's been a bit of talk about renaming the lane, renaming Crossley Lane, I think that's a fantastic idea and I would fully support that," Mr Andrews told reporters.

The espresso bar Mr Malaspina co-owned since 1974 is expected to reopen in a limited capacity on Tuesday.

His family plan to donate the floral tributes to nursing homes and hospitals to brighten others' lives, just as Mr Malaspina did.

Australian Associated Press