Words race out of Eusebio Di Francesco’s mouth like cars entering il Gra, the orbital motorway that encircles the city of Rome. They are going a mile a minute, bumper to bumper, yet the man behind the wheel somehow finds a way to transmit an overwhelming sense of calm.

Roma’s manager speaks fast because this is no time to drop a gear following his team’s return to their Trigoria training base after a vital league win away at Spal. They have barely one day here before they pack up again and fly to Liverpool for the first leg of their Champions League semi-final.

“This season has been one of great growth for us,” he says, eyes bright in defiance of the bags beneath them. “In recent years Roma had not managed to compete through to the latter stages of multiple competitions. We are still competing [for a top-four spot] in Serie A, where Liverpool have already secured a bit more certainty that they will be back in the Champions League next year.”

Roma began this weekend third, just a point clear of Internazionale in fifth. The game against Liverpool will be their eighth in 25 days, a stretch that has included not only a two-legged quarter-final with Barcelona but also a draining derby against a Lazio side fighting to leapfrog them in the table.

If this team is running close to empty, you would not know it from the grins on players’ faces as they arrive for training. “What we do is a job but it needs to be fun, too,” says Di Francesco. “It should be a joy. That’s what I always say to the lads. The first thought is to prepare to enjoy ourselves together, and work hard. They have to enjoy themselves. This is a game first of all.”

That lesson was learned during Di Francesco’s own playing career. As a member of the Roma team that won the Scudetto in 2001 – only the third in their club’s history – what he remembers above all is the way that he and his team-mates used to spend time together off the pitch, hanging out to play cards or shoot pool.

He does not see great similarities between that team and this one. “Because these are completely different times. There’s a greater professionalism now, as there should be, but back then there was more of a family spirit. That’s been lost a little bit, with social media, with technology, with the arrival of new ways of working. If those two things could come together, that would be the ideal. And we are trying here to make it so.”

That he thinks a great deal about players’ psychology is obvious. After Roma overturned a three-goal deficit in the second leg of their quarter-final against Barcelona, pundits rushed to acclaim his tactical nous, switching for the first time to a three-man defence: some move for a man who has always insisted that 4-3-3 is the “ideal formation”.

Yet Di Francesco frames the change differently. “Football is dynamic,” he asserts. “Even when you talk about a four-man defence, you often end up defending as a three, or even a two, depending on the game situation.

“My decision to change the system was linked to the fact that with some teams, with the characteristics of certain players, a three-man defence can give you a little bit of extra physicality. You get a little bit of extra strength – some of that just in the heads of the players themselves. Sometimes, especially in Europe, you need a little bit more physicality.”

That was certainly the case against Barcelona, bullied into submission by Roma’s muscular midfield trio of Radja Nainggolan, Kevin Strootman and Daniele De Rossi. Does Di Francesco plan to repeat the ruse against Liverpool? He would hardly tell us if he did, yet he does observe that Jürgen Klopp’s team are a long way from the old English stereotype of long balls and reducers.

Appointed last summer, Di Francesco never had a chance to work with Mohamed Salah. If there is any regret at missing out on the chance to coach such a talent, then he hides it well, though he does note his players’ praise for “a great guy, a great professional, a hard worker”.

“I don’t need them to tell me,” he says with a smile when asked if he has sought out advice from his defenders on how best to frustrate Salah. “His qualities are very clear. Don’t forget that I prepared games against him in Italy, too. But the fact lots of our players know him well, that can be an advantage.”

Salah has scored 41 goals this season, yet it was his assists last year at Roma that helped Edin Dzeko to reach 39. The Bosnian has a more modest 20 this time around, and Di Francesco offers frankly that “he has alternated good matches with not-so-good ones”. The good, though, have been very good indeed. He notched a double at Stamford Bridge and scored in both legs against Barcelona.

Does Di Francesco think Manchester City made a mistake in letting go of a player so capable of taking the biggest games in hand? “It seems to me Guardiola loves a different type of forward. One who’s a little more mobile, a little faster. Edin is a fantastic player, but with different physical and technical attributes to the ones Guardiola wants.”

What Di Francesco seeks most in his own players is a desire to be part of something. Asked whether he is grateful to have a pair of Romans in his team – De Rossi and Alessandro Florenzi – he replies with a “yes”. “But I think every player should have that sense of belonging to the place where they play. Not just those guys. When I was at Sassuolo, Sassuolo were the most beautiful team in the world, the best, I was black and green [the colours of their shirt]. Now I’m yellow and red, not out of hypocrisy, but because we all need to have that sense of belonging in our work. Every person needs to feel that passion and the love for their team.”

He is banging the table now, swept away with the force of his own sentiment. Italy’s biggest clubs first came calling for him in the summer of 2016, yet he turned them down then because he could not stand to walk away from Sassuolo in a year when he had just led them to their first ever Europa League qualification.

Di Francesco believes in seeing a project through to its conclusion. For Roma, he hopes, a first Champions League semi-final in 34 years is just the beginning.