Agent Giuseppe Riso, who represents both defender Mancini and midfielder Cristante among other clients, met sporting director Tony D'Amico to finalise the renewals.
According to Gazzetta dello Sport, the two extensions will be announced shortly, pleasing head coach Gian Piero Gasperini, who had asked the club to secure his key players for next season.
The negotiation was not as straightforward as it once appeared.
Under previous management, terms seemed close to being agreed and only needed to be finalised, before talks stalled abruptly and put the situation back in doubt, with Inter attempting to move in on the Italian defender.
In the end, both players will remain in the capital.
Mancini has made 319 appearances and scored 23 goals for Roma, adding 48 appearances and seven goals at Atalanta and 30 appearances and one goal at Perugia earlier in his career.
Cristante has made 364 appearances and scored 21 goals for Roma, with 59 appearances and 15 goals at Atalanta, 20 appearances and one goal at Benfica, 18 appearances at Pescara, five appearances and one goal at Milan and four appearances at Palermo.
Mancini has won 20 caps for Italy, scoring twice, while Cristante has been capped 48 times, also scoring two goals.
Elsewhere, Dovbyk reflected on his time at Roma in a lengthy interview with Ukrainian YouTube channel Rosso Dritto, touching on his failed move to Atletico Madrid and doubts over his future in the capital.
The striker explained that his decision to join AS Roma stemmed from his relationship with former head coach Daniele De Rossi.
"With Atletico everything was done, then suddenly the answers stopped coming. De Rossi called me and there was an instant connection between us.
"I gave him my word and I kept it, even when Atletico came back in. I never imagined he would be sacked after just three matches. After Totti, he's the biggest legend at Roma, it felt surreal."
Dovbyk also recalled the difficulties of a season played under three different coaches.
"We lost so many matches, the fans whistled at us in front of 70,000 people. Then Ranieri arrived, a very wise man. Despite everything, I scored 17 goals."
His verdict on the present campaign was blunt.
"For me this season has been a failure, partly because of the injuries. In the summer I was close to Milan, then in January West Ham were there, but I wanted to finish the season at Roma. Now I want to understand if there's still a place for me here."
He also offered a thought on coach Gasperini.
"I think it will be the toughest training of my career. Malinovskyi reassured me: 'Nobody has died, so it will all be fine.'"
On playing at the Stadio Olimpico, Dovbyk added: "The atmosphere is incredible, football is lived to the full here. In Rome you're either a king or you fall."