Carlton want interim coach Josh Fraser to pitch for the full-time head coaching job.
Fraser has been given a week to decide if he wants to bid for the role, with the Blues expecting to announce their next permanent coach by the end of August.
Fraser has steadfastly ruled himself out of vying for the position despite reviving Carlton's fortunes with eight wins and two losses since replacing Michael Voss.
"We have moved into the second phase of our process and we have now asked Josh to be part of that," Carlton's football manager Chris Davies told SEN radio on Thursday.
"It's fair and reasonable to say that after 10 weeks, he has a better understanding of where he's at.
"I have had a conversation throughout this week to say that we think he should put his hand up. We have given him the next week or so to think about that."
Current assistant coaches Jaymie Graham (Fremantle), Corey Enright (St Kilda) and James Kelly (Geelong) are understood to be among the candidates.
"I'm going to be careful in not confirming who's in or out of our process," Davies said.
"What I will say is, ultimately, the best person for Carlton is the person who wants to do the job as well as getting through our process.
"I have got no issue with Essendon, or when Tassie were going through their stuff, or feel no pressure from anyone else going through the process at the moment."
Davies expected a decision by the end of the home-and-away season, which concludes on August 23.
"Given we're into the next phase of the process and there are four games to go in the minor rounds, I'd be surprised if we're not in a position to be dealing with it before the end of that period," he said.