Nasri, who has just one year remaining on his contract at Arsenal, has long been tipped to complete a move to Eastlands and The Times even reported it as a done deal over a month ago.
With Arsenal yet to accept City's offer for Nasri, though, Mancini has been left frustrated at the board's frustration to conclude their transfer business as he prepares for the opening game of the season against Swansea on Monday.
"Last year was the same too," he said. "I don't understand this. I thought we could get all the players we needed three or four weeks ago, but here we are and the deals aren't closed.
"We need this player [Nasri]. I asked for this player two months ago. I'm worried because I don't have this player today and I probably won't have him tomorrow. We play three games in August and I am worried.
"If we complete our squad, we can fight for the title this year, but that is a problem at the moment because we play our first match [on Monday]."
He added: "It is not important whether you are the richest club or not. The market is difficult - I understand that - but when you have your targets, it is important to go very hard.
"It was important to have these players for pre-season because it is there that you prepare for the season. At the moment, we don't have these players. I asked and I hope we can have them this week."
Fiorentina winger Alessio Cerci is among the other players Mancini is looking to sign and he said: "Cerci is a good player. He is young and he could be an option but there are other players."
City have already signed Sergio Aguero, Gael Clichy, Stefan Savic and back-up goalkeeper Costel Pantilimon this summer.