But his comment was disputed by both May and Clarkson on Twitter.
When Clarkson, at the wheel of an orange Aston Martin, was asked whether he had a message for Evans, he said, "don't be stupid" and tried to grab the camera.
The trio were mobbed as they drove into the city's Piazza Dei Signori.
Clarkson had invited fans to join them when the day before when he tweeted: "People of Italy James Richard and I will be filming in the Piazza Dei Signori in Vicenza tomorrow at 3pm. Come along."
Crew working on their show claim Clarkson intentionally staged the scenes of chaos in the stunning piazza as "a two fingers up" to his TV rival, Evans.
"Jeremy's been trying to keep quiet about it, but he's pleased as punch," The Daily Mail reported one unnamed cameraman as saying.
"He's kept his mouth shut about Evans the whole time, and now Evans has hung himself by his own rope without Jeremy getting involved.
"It's ironic because Evans was a massive Top Gear fan, but he's ruined it for himself by taking the top job job on the show."
The cameraman spoke as thousands of adoring fans mobbed the trio as they drove supercars slowly through the ancient Piazza Dei Signori in the centre of Vicenza, northern Italy.
The cameraman told MailOnline that Clarkson revelled in the crowds - adding: "Jeremy intentionally created this mob by sending a tweet out," he said.
Another crew member added that the filming in the Piazza Dei Signori was a "victory lap" to celebrate Evans' fall from grace.
Thousands of young Italians gathered in the piazza from 3pm, but the former Top Gear presenters did not arrive until more than an hour later.
Most of the fans were young men, but there was a significant number of teenage girls and young women in the crowd.
Programme bosses have said the Grand Tour studio will be a big tent which will be taken to a different location each week to record a series of 12 programmes with a live audience.
The hosts will then film pre-recorded segments reviewing cars and completing challenges, as they used to do with Top Gear.
The name of the programme is a reference to 'The Grand Tour' undertaken by 17th Century Englishmen such as Lord Byron when they visited France, Italy and the rest of Europe in search of cultural edification and personal enlightenment.
Its initials will be GT, an inversion of the TG nickname given to Top Gear, highlighting the rivalry that has grown since Clarkson was dumped by the BBC.
