Married couple Roger and Wendy Bannister, from Stourbridge in the West Midlands, said they received the invitation from mayor Gianni Alemanno.
The move comes after Mr Bannister, 58, and Mrs Bannister, 61, along with Mr Bannister's brother Stephen, 68, and his wife Joyce, 69, were landed with the bill on Sunday, the last day of their holiday.
The group had stopped at the Antica Roma ice cream parlour and bar close to the Spanish Steps in the city centre and ordered three scoops of ice cream each. They did not ask to sit down but to take the ice creams away.
Mr Bannister said he reacted with disbelief and shock when he received the bill of 64 euro, or £54.
"It was just disbelief and then giving the money and having the change slapped down, the five euro note and the one euro coin, and then the receipt - 64 euro - we just walked out in disbelief," he told BBC Breakfast. "It was shocking really."
The incident has been widely reported in the Italian media.
He said they had received a message expressing "sincere apologies", saying Mr Alemanno, an elected politician who is accountable for the strategic government of Rome, was "very upset" about what happened.
Mr Bannister said they were told they would be treated to an ice cream if they returned to the city. He added that they were considering the invitation.
A unnamed staff member at the ice cream parlour told the Daily Mail: "The prices are clearly on display. They got what they asked for - four ice creams."