Chris Wilder, the United manager, admitted earlier this week that he wants to sign a new midfielder before the transfer window closes in less than 48 hours time after identifying a vacancy within his squad.
Although Phillips is yet to be the subject of an official approach from Bramall Lane, The Star can confirm United's scouts added his name to a list of potential targets after they pipped Marcelo Bielsa's side to automatic promotion last term.
With United's arch-rivals losing in the play-off semi-finals to Derby County, the 23-year-old has yet to agree a new contract at Elland Road despite entering the final 12 months of his present deal.
Aston Villa, who eventually joined United and Norwich City in the top-flight, have also been linked with Phillips. While the player will be attracted by the possibility of plying his trade at the highest level, Elland Road's board of directors are attempting to effectively price him out of a move elsewhere.
Speaking ahead of this weekend's visit to the Vitality Stadium - which marks United's return to PL competition following a 12 year absence - Wilder acknowledged he still wants to recruit a 'box to box' player despite welcoming eight new faces to South Yorkshire since guiding his team out of the Championship.
"Yes, we're still hoping to do a little bit more," he said. "We're out there, looking and trying. You can never say for definite, that's impossible in this business, but we're confident we can get something across the line and sorted."
"We've done a lot up front," Wilder added, after completing a £20m swoop for Oli McBurnie, added. "We're okay there I feel, and we're very pleased with what we've done."
United, who are due to receive a small percentage of the £80m fee Steelphalt Academy graduate Harry Maguire commanded when he joined Manchester United from Leicester City this week, could be in line for another unexpected windfall if Ben Whiteman swaps Doncaster Rovers for Hull City.
The youngster, who also progressed through United's youth system, has attracted a bid of around £2m from the East Yorkshire club.