Producers are said to have approached Barbet in the hope of giving the ITV breakfast show a greater focus on current affairs, following the departure of Chiles and Christine Bleakley.
Daybreak bosses are apparently planning to pair him with BBC Breakfast host Sian Williams, who was offered the job earlier this week.
Current support hosts Dan Lobb and Kate Garraway will present Daybreak until the new duo arrives in the New Year, according to the Sunday Mirror.
"Matt and Sian are seen as the ideal team to bring more of a newsy feel to Daybreak," a source told the paper.
"It is a bit of a gamble as ITV cannot afford another high-profile failure, they are pulling out all the stops to land them as presenters."
Newsreader Natasha Kaplinsky was previously said to have ruled herself out of the running to become the new Daybreak host.