Despite the rumours previously being denied by team principal Maurizio Arrivabene, the paper reports that they could in fact be true after Ferrari missed their pre-season target of wins early on in the season.
Allison, whose wife died earlier this year, has been commuting to Italy from the UK in order to spend more time with his children but a full-time move back to his homeland with Renault has been mooted.
Ferrari has apparently singled out former technical director Ross Brawn as the man to replace Allison and turn the team's fortunes around.
But the Brit, who has had championship success with Benetton, Ferrari, Brawn GP, and was instrumental in laying the foundations for Mercedes' current success before leaving the Anglo-German team at the end of 2013, has insisted that a return to F1 does not appeal to him.
Speaking at the Goodwood Festival of Speed last month, Ross Brawn said that he currently has no motivation to return to the sport full time.
"Never say never about any of those sort of things, but I'm quite content doing what I'm doing and nothing has come along that I would be motivated and interested in," he told Martin Brundle at the event. "My nature when I was involved was 24/7 and I'm not sure I want a 24/7 any more."
La Gazetta's Luigi Perna also played down the speculation, saying the promotion of chief designer Simone Resta or the signing of Toro Rosso's James Key was more likely.
"Brawn recently declined offers to return to F1 and would not be willing for such a demanding role," he said. "Not to mention that F1 has changed in the several years he has been outside of the sport."
Bahrain International Circuit - Winter testing