The 25-year-old, who can also operate on the left side of midfield, will move to Carrow Road on a four-year deal from the Championship club for an undisclosed fee once all of the relevant paperwork has been completed.
Olsson has been a long-term target of Canaries manager Chris Hughton, who has been on a summer spending spree following the club's record £8.5million capture of Holland international Ricky van Wolfswinkel, while England Under-21 winger Nathan Redmond was last week purchased from Birmingham, and defender Javier Garrido has signed permanently after a loan from Lazio.
"We are delighted with the addition of Martin. He's been playing in England for seven years and is well acclimatised to the Premier League," said Hughton on the club's official website.
"He can also play anywhere on the left hand side so he brings us good flexibility in that sense, plus at 25 he's at an age where he has some experience and also some time to improve.
"At this time it is about improving the squad that we have and Martin certainly does that."
Around an hour later, the club confirmed the arrival of Nash on a one-year deal following his release from Stoke.
"It is now important to us to bring in a keeper of great experience and a player that is used to living life in the Premier League," Hughton said.
"The move also allows us to send Declan Rudd out on an extremely valuable season-long loan to Preston where he can get some first-team football under his belt."