"He will be gaining the attention of an awful lot of people, and quite rightly so. We want him to stay here," Mills told BBC Sport.
"Mark Ashton (Ipswich chief executive) and his team did a tremendous job when they went out to find him - I certainly didn't know who he was. When I heard Kieran McKenna got the job at Portman Road, I did have to ask the question: 'Who's Kieran McKenna?'
"This is not just about Kieran, it's about the club as well - it's a big club with big opportunities, the ownership is good, the off-the-field team is good, so I dearly hope he stays here for a long time. But I think he can go right to the very top, I firmly believe that.
"It worries me to death that Manchester United haven't made a decision (about who will be in charge there next season)."