The England Under-21 goalkeeper, who recently signed a new contract with the Reds, played on loan at Shrewsbury last term.
Speaking on Manchester United's club website, the 21-year-old said: "I can't wait to go there to build on the last season I had at Shrewsbury.
"It's in the division above and showing that progression by going on to the next bigger stage, in front of 30,000 passionate fans at Bramall Lane.
"I can't wait to go there and perform and connect with them.
"I'm sure it will be a successful season.
"Sheffield United are a massive club and it's a massive city for football.
"They've got that rivalry with Sheffield Wednesday and I'm looking forward to those games coming up as well.
"I can't wait to get there and push up the division."
After an impressive season with Shrewsbury, the goalkeeper was rewarded with a new contract at Old Trafford.
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Henderson is following the paths of fellow United stars Joe Riley and James Wilson, who have both spent stints with the Blades.
He was named in the PFA team of the season for League One and has attracted interest from a host of clubs.
With Sergio Romero and David De Gea blocking his path to the first team, the 21-year-old will be getting the experience that he needs with the Chris Wilder's side.
Sheffield United finished tenth last season, with 20 wins from 46 matches.
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