Weir was dismissed from Bramall Lane on Friday night due to the Blades' poor start to the League One season.
United are currently in the relegation zone with one win from their first 10 games.
"The decision to make a managerial change was not taken without some very sound thinking about what the alternatives might be," Phipps is quoted as saying by Sky Sports News.
"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 Derby County boss Nigel Clough is the early favourite to be appointed the club's new manager.