Weir was sacked after a dismal start to United's third consecutive season in League One, winning just once in 13 league and cup games.
The rumour mill has well and truly kicked into gear, with the likes of Nigel Clough, Tony Pulis and Leyton Orient's Russell Slade all being linked with the vacancy.
Phipps, however, wouldn't be drawn on naming any individual names, but did say an announcement is likely next week.
"I think I'm reasonably certain (that a new manager will be installed before next Saturday's league game at home to Port Vale)," said Phipps.
"We are not saying anything about who is on the shortlist, but we have the advantage of there being a number of excellent options, which might not always be the case at this point in the season.
"I've received a lot of unsolicited advice from people who are very hopeful that we'll make the right decision; I think they'll be gratified.
"We have got a shortlist, but I'll let the length of that shortlist remain a mystery though. We hope for a relatively speedy adjustment and we'll be back to a permanent solution quickly.
"We'll have more to say on this, hopefully, in a matter of days - we are pretty clear about what our best options are.
"Not everyone," was Phipps' reply when asked if the club had made contact with all parties that interested them. "We have had indirect enquires made but our best guess is that we're on the path to having a very suitable candidate to lead the way."
Phipps joined the Bramall Lane board in September when the deal for Prince Abdullah bin Mosaad bin Abdulaziz Al Saud to become co-owner of the football club with Kevin McCabe was announced - Phipps is the Prince's principal advisor.
The Blades travel to Northampton on Sunday afternoon to face Coventry City under the temporary leadership of Chris Morgan. Morgan will be assisted by goalkeeping coach Darren Ward and head of academy coaching Travis Binnion for the live TV game, kick-off is at 4:00pm.