The Dons could not halt Ronny Deila’s team last season but kept their title challenge alive until the end of the campaign and recorded their highest finish in the top flight since 1994.
Losses to Celtic in all four of their Premiership meetings effectively ended their charge but Miller, who made nearly 800 appearances for Aberdeen, winning three Scottish Premier Division titles along the way, said they can challenge again this year.
He told Sky Sports News HQ: “Aberdeen made a valiant effort of it last season. They kept it going until April and fell away towards the end of the season.
“They have strengthened their squad as well, are playing some good football and have got a lot of experience. If you are looking around the game then the challenge should come from Aberdeen.
“Whether they can lift themselves to overcome Celtic is another matter but I think they will make it difficult again this season.”
Aberdeen have had to undertake a six-hour flight to face Kairat Almaty of Kazakhstan in their Europa League third qualifying round first leg on Thursday.
And Miller, who captained the side to the European Cup Winners’ Cup in 1983, has accepted that a place in the group stages of the Europa League will stretch McInnes’ options.
“Absolutely, I think that will test his squad but Derek has strengthened there,” Miller added. “He has brought in more experience and that will have an effect.
“You want to be competing at the highest level and in Europe and try to keep the league campaign going at the same time.”
Watch Celtic v Ross County live on Sky Sports 1 from noon on Saturday, August 1 and Dundee United v Aberdeen also live on Sky Sports 1 from 3pm on Sunday, August 2.