United wanted £200,000 as compensation but both clubs have agreed that a friendly will be played at Bramall Lane while a un-named Leeds player will join the Blades on loan.
Reid said: "We have tied up most things, but I'm delighted to now have him on board. Contrary to all reports, I know [Sheffield United chief executive] Terry Robinson really well and there has not been a cross word between us. People have to realise that sometimes these things take a while to sort out, but he is now coming in as head coach.
"I decided to make the choice myself and although I didn't know him personally, I did know of him and his reputation for working with players.
"I know he is delighted to be joining us and he is now going to have a big responsibility working with the players here at Leeds."