The Blades axed rookie boss David Weir on Friday after just 123 days in charge following a dismal run that had left the club in the League One drop zone and without a win since the opening day of the season.
Former Derby manager Nigel Clough has quickly emerged as the frontrunner for the Bramall Lane hotseat, with ex-Stoke chief Tony Pulis, Leyton Orient's Russell Slade and Crawley's Ritchie Barker also linked with the post.
Phipps, who joined the United board last month in his role as principal advisor to the club's new co-owner, Saudi Arabia's Prince Abdullah, admitted Weir's early exit was lamentable - but said progress had already been made over a successor.
"The decision to make a managerial change was not taken without some very sound thinking about what the alternatives might be," Phipps said.
"We have a shortlist and we have made some progress on our shortlist. We expect to take some action in a matter of days rather than weeks.
"It would be unfair to say who's on that list but I doubt anyone on there would be a surprise."
Former Blades defender Chris Morgan, who stepped in when Danny Wilson was sacked towards the end of last season, will take charge of Sunday's trip to Coventry - live on Sky Sports - but Phipps said a new man could be at the helm by the time Port Vale visit on 19 October.
"We are moving as quickly as possible," Phipps added. "It's a plausible outcome to have someone in place for the 19th so we've announced that Chris will be covering just for this Sunday."
Phipps admitted it had been a tough decision to cut loose a man given a three-year contract this summer but suggested the former Everton and Scotland defender had tried to implement change too swiftly.
"It's a very difficult thing to change a manager; one brought in to be a long-term solution and a guy who couldn't be classier," Phipps said.
"But we find ourselves in a a place where it's unavoidable. Maybe with another four or five points we would have been able to delay longer and there would have been time for things to click but we weren't making the pace.
"We don't take the view that our problems down to one person but maybe we didn't keep an eye on the pace of change. It was maybe too much, too fast. With lots of new players, we're the least settled team in League One
"I respect David Weir for having the courage of his convictions. It didn't work. I only wish that the structure of English football was less brutal but that relegation threat is a very significant threat."
Watch Coventry v Sheffield United live on Sky Sports 1HD from 3.30pm on Sunday.