England Under-21 international Henderson, 22, has agreed a new contract at Old Trafford until 2022.
Osborn, 24, has signed a three-year contract at Bramall Lane.
He came through Forest’s youth set-up and, after making his debut in 2014, played 230 games for the Championship club.
Blades boss Chris Wilder said: “He’s been a fantastic player for Forest in the Championship and I know for certain that two or three clubs in January were sniffing around to take him.”
Wilder also said Henderson, who kept 21 clean sheets last season, was “always our number one target”.
Sheffield United finished as Championship runners-up to Norwich City last term.
They have now made seven summer signings, having already recruited forwards Lys Mousset and Callum Robinson, midfielders Ravel Morrison and Luke Freeman, as well as re-signing veteran defender Phil Jagielka.
Find all the latest football transfers on our dedicated page.