Free agent Nielsen, from the Faroe Islands, has previously had spells with Blackburn Rovers and Motherwell on loan.
Nielsen has penned a two-and-a-half-year deal with City and joins fellow shot-stoppers Shay Given, Joe Hart and Kasper Schmeichel at Eastlands.
Meanwhile, City boss Mark Hughes has stressed his desire to reach the last 16 of the Uefa Cup.
Hughes' men drew their first leg clash at FC Copenhagen 2-2, and the Welshman has confirmed he intends to field a strong starting XI for the return fixture on Thursday.
He said: "When you are in Europe and you are playing Thursday and then Sunday in the Premier League sometimes you think it is more trouble than it is worth.
"But when you actually go out of a competition such as the UEFA Cup then there is a gap in your programme and a feeling of loss.
"I do not want that to happen at City, especially as they have not made that many excursions into Europe.
"From a development point of view we need to stay in the competition as long as we can."