Henderson's loan is due to expire on June 30, with any extension for him to remain at Bramall Lane for the rest of the 2019-20 campaign, which now ends on July 26, needing to be finalised by Tuesday.
"I believe it's now just administration with the two clubs," said Wilder. "I had a chat with Ole and he was fine with extending his loan until the end of the season.
"Hopefully, that will get signed off before the deadline. I appreciate Ole's help on that one. Dean has been great for us and we have been great for Dean."
Under the terms of his loan, Henderson is ineligible to play against his parent club when the Blades travel to Old Trafford on Wednesday evening.
"I have made a decision about who will play in goal. I came to that decision when I knew Dean wasn't going to be available, but I'm not willing to share who will play," the Blades boss said.
"Why would I share it? What positive do I get out of telling you my team?"