The Blades reached an agreement with Arsenal to sell Aaron Ramsdale to the Gunners earlier in the transfer window and as deadline day dawned wanted to snap up another shot-stopper.
They explored a host of options as they looked to make a signing and prepared a £5m offer for Rajkovic.
The Serbia international custodian is on the books at French club Reims and Sheffield United were ready to launch a swoop for him.
However, they received the green light to land Olsen from Roma; the Swede was their primary target.
Olsen, who spent last term on loan at Everton, has landed from Roma on a season-long loan deal.
The 31-year-old will be looking to play regularly at Bramall Lane and help Sheffield United to win promotion back to the Premier League this term.
Sheffield United are currently second bottom in the Championship after collecting just two points from their opening five games.
