The Belgian limped off after just seven minutes of the 1-0 last-16 first-leg loss at the Jose Alvalade Stadium and will definitely miss the Barclays Premier League leaders' trip to Swansea on Sunday.
"I hope that for Vinnie, the injury will be short," Mancini said. "When you have a calf (injury), it can be a big problem. I hope he can recover in 10 days, two weeks maximum. We will see how it is in another one or two days."
Joleon Lescott, who replaced him, also felt pain in his groin during the game but manager Roberto Mancini does not believe that problem is serious.
On the positive side right-back Micah Richards, again left out to rest a minor knock, should be fit as City look to protect their two-point lead at the top of the table at the Liberty Stadium.
Mancini said: "I didn't take a risk with Micah because Pablo Zabaleta is injured. Micah will play on Sunday."
City are now staying in Lisbon to train in warm weather for the next two days before travelling directly to south Wales.
City will now have to overcome a deficit to reach the Europa League quarter-finals after Brazilian defender Xandao scored the only goal after 51 minutes.
Mancini said: "It wasn't a surprise because anything can happen in football, but I don't think we deserve to lose. We didn't play very well but could've scored, then conceded a goal, so that is it. We have another 90 minutes and I think it will be a different game. We still have a big chance. Hopefully we will have a chance to score and I think we will score.
"I don't think they had a lot of chances to score. Sporting had players behind the ball always, to play on the counter-attack. They had players behind the ball for this reason and I think we had more chances than them."