The Dons have been sparring with the Parkhead club this season and can move level if they win Sunday's match at Celtic Park.
Mulgrew will miss the title showdown due to injury and thinks his team-mates will have to respect his former club.
He said: "We will need to be aware that they are very tight at the back but they do have the threat going forward. They have goalscorers in their team."
Mulgrew also hinted that many people south of Stirling don't realise that Aberdeen is a big club that should be taken seriously as potential Scottish Premiership champions.
He added: "They've obviously won the league in the past and European cups and when you are up there you realise how big a club it is.
"They are probably where they should be and they will be looking to challenge us all the way. But it's up to us to see off that."
Source: The Herald