Mancini has said he is also hopeful of securing the signatures of Inter Milan full-back Maicon, Fiorentina defender Matija Nastasic and Benfica's defensive midfielder Javi Garcia before the transfer deadline.
On what is set to be a busy day of activity for the Barclays Premier League champions, Mancini is also expecting midfielder Nigel de Jong to join AC Milan.
The move for former England, Arsenal and Everton keeper Wright has come as a surprise. The 34-year-old joined League One Preston earlier in the summer but left citing homesickness. Mancini said: "We need a goalkeeper with experience and Richard Wright is good for us."
City have been linked with Sinclair for most of the summer and have agreed a fee of ?6.2million, rising to a possible ?8million, with the Swans. Mancini said: "We are working, we are waiting. I think we have a good chance to close but we need to wait."
Brazil international Maicon, 31, played under Mancini at Inter but is now in the final year of his contract, while Garcia, 25, is highly rated and would cost a reported ?25million. Nastasic, 19, would bolster defensive ranks with Stefan Savic possibly heading out of the Etihad Stadium.
Mancini said: "We have another 10, 12 hours to close on these players. We need more time. I am confident. I think if we close these players, I think we improve our team.
"They come to play in a different championship, they need time. We are talking about one top player in Maicon, because he has experience. Nastasic will be one of the top players for a long time."
The likely departure of De Jong is not a surprise. The 27-year-old Holland international is in the final year of his contract and has been linked with other clubs throughout the summer.
Mancini said: "I am sorry because Nigel was an important player for us but if he is happy, I am happy for him. I am sure he will do really well in the Italian championship."
Source: PA
Source: PA