Adkins made the prediction after Saturday's clash between his old and current club, as United beat Hull City 1-0 at Bramall Lane thanks to David McGoldrick's second-half penalty.
The victory sent United top of the Championship table going into the international break, and Adkins was left in little doubt that the Blades are top-flight bound under Wilder.
'We had three close games,' said Adkins.
'Middlesbrough who we drew with, Leeds United on Tuesday, and Sheffield United. They are three teams who I think will get promoted this season. Three good sides.
'Why those three? The three teams all play different ways and it shows there's different ways to play football and win games. Middlesbrough are well equipped in what they go and do. 'Leeds, whether they can keep everyone fit is the other side of it, but some of the football they play is really good.
'And Sheffield United are now well equipped. They have an excellent manager, an excellent group of players and are going to be able to deal with injuries.
'They have got a real passionate support who will get behind them.' Adkins lasted one season in charge at Bramall Lane, until the Blades slumped to their lowest league finish since 1983 and he was sacked. Wilder took over and led United to the League One title in his first season, before they finished a respectable tenth in the Championship after being in contention for the play-offs until the penultimate game of last season.
'Sheffield United have suffered over the past few years, but they are on the up now and it's only a matter of time before they are back in the Premier League,' Adkins added.
'If you flip it the other way, Hull City, having been in the Premier League, are now going through that transitional period where you have to get all your finances right, the stability of ownership right, and a group of players and team with a way of playing.
'That takes time and we are going through that transition now.
'Sheffield United have come through that and I think there's only positive things going to come for the club in the future.'
Defeat at Bramall Lane sent Hull bottom of the league, leading to inevitable questions over Adkins' future at the club.
'Everybody understands what football is,' he said.
'People want change, they constantly want change. I am long enough in the tooth to understand what it is all about. But what I can see is a group of players who are grafting and working really hard. They have demonstrated that this week against teams who wereÂ first, second and third in the division.
'We have drawn 1-1, been beaten 1-0 from a 30-yard shot and 1-0 by a dubious penalty. We have done quite well in all three games. We will find out about ourselves, our team-mates and the people we surround ourselves with.
'What we have is a unity and a group of people who will fight for the cause. But the fact of the matter is we are where we are in the division. We can't hide away from that. It's not where we want to be.'