After a season of remarkable inconsistency, Italian media have reported that Di Francesco's job will be at stake when they defend a 2-1 lead away to the Portuguese champions at the Estadio do Dragao in Wednesday's second leg.
Former Fiorentina coach Paulo Sousa is the favourite to replace Di Francesco should he be fired, according to Italian media, who said the Portuguese will be in the stands on Wednesday.
Another possibility for a replacement is former Roma defender Christian Panucci, who is currently coach of the Albanian national side.
The club has not made any official comment on the coach's situation.
Roma had just got over their 7-1 Coppa Italia drubbing by Fiorentina in January when they were thumped 3-0 by neighbours Lazio in their derby on Saturday, further stretching the patience of the fans and the club directors.
Di Francesco himself admitted it was not good enough and was especially angry at Lazio's first goal, scored from a move which began with a throw-in.
"You can't concede a goal through the middle from a throw-in – it was all too easy for them. We got our marking wrong," he said. "It can happen to anyone but it's been happening a bit too often to us this season.
"We really must work on the mental side of things. We seem to be on and off."
The 49-year-old led Roma to third place in Serie A and the Champions League semi-finals last season - his first at the club - was rewarded with a new two-year contract.
Although there have been some impressive performances this season, especially from 19-year-old midfielder Nicolo Zaniolo, they have been overshadowed by Roma's tendency to collapse.
Roma have been beaten by Bologna, SPAL and Udinese - all teams in the lower half of the table - and also managed to blow a two-goal lead at home to bottom-of-the-table Chievo, when they were held 2-2.
Then, in the space of four days in January, they squandered a three-goal lead at Atalanta to be held 3-3 before suffering their seven-goal mauling at the hands of Fiorentina.
Di Francesco has been left fishing for explanations and there have been some unusual ones, such as the suggestion he made earlier in the season that striker Edin Dzeko is less effective in daylight.
"It seems he prefers playing at night," said Di Francesco.
(Writing by Brian Homewood; editing by Sudipto Ganguly)
