The reigning champions avoided defeat at Anfield after the Reds gave away the lead twice in a 2-2 thriller on Sunday. Mancini's men came back to win 3-2 at home against Southampton in their opening fixture.
Everton midfielder Jack Rodwell joined the Citizens - their only signing this summer while Swansea City winger Scott Sinclair is expected to complete a £6.2 million to Eastlands soon.
However, Mancini wants a high-profile signing prior to the Transfer deadline this month end. "We need to bring players in and we have five days in which to do it," he said.
"We have to work hard in these four or five days. It will be a long week for us. Yes, we have our targets and we will go for them in the next five days," he added.
Mancini has been targetting Atletico Bilbao midfielder Javi Martinez and Stefan Jovetic of Fiorentina while he has not given up hopes of signing Roma's Daniele De Rossi. City have also been linked with Inter Milan right-back Maicon.
City bosses had reportedly asked Mancini to offload the line of strikers namely Edin Dzeko, who has been linked with Juventus, while the champions rejected a bumper offer for striker Mario Balotelli.