Greenwood's move to Turkey is understood to be all but agreed, with the Super Lig club offering a lucrative contract Roma were unwilling to match.
That leaves D'Amico focused on other targets, with Dan Ndoye - a familiar face in Serie A from his time at Bologna, now at Nottingham Forest - already known to be one of two priorities identified by the sporting director, according to La Gazzetta dello Sport.
The other is Can Uzun, the 20-year-old Eintracht Frankfurt playmaker who has also drawn interest from elsewhere in Italy, with AC Milan and Napoli previously mentioned as admirers.
Eintracht Frankfurt have set his valuation at €45m, a figure AS Roma consider too high, though there is understood to be some room to negotiate a more workable fee. Uzun, who holds both Turkish and German nationality, has featured for Turkey at the World Cup, appearing as a substitute against Paraguay and the United States.
D'Amico is expected to push hard on both fronts as he looks to strengthen Gian Piero Gasperini's attacking options this summer, with Roma keen to avoid the kind of prolonged, high-cost pursuit that ultimately cost them Greenwood.
Uzun has emerged as one of the Bundesliga's most exciting young talents over the past two seasons, scoring in each of Frankfurt's first five league matches last term before a run of injuries disrupted his momentum.
He finished the campaign with seven goals and four assists in 21 Bundesliga appearances, having previously contributed two goals in five Champions League outings the season before.
D'Amico has already opened dialogue with Uzun's representatives, who also look after the interests of another player being monitored by Roma, Nicolo Tresoldi, though any deal for either Ndoye or Uzun is unlikely to move quickly given the contract renewals - for Paulo Dybala, Zeki Celik and Lorenzo Pellegrini among others - currently taking priority at the club.