The Demons are keen on securing Dawes, who has told Melbourne and two other clubs, believed to be Carlton and Western Bulldogs, that he will make a decision by the end of the weekend.
Dawes, who played under Neeld when he was an assistant coach at Collingwood, is considering a change after former West Coast big man Quinten Lynch joined the Magpies to play a similar role.
"We've been told we're one of three, so at the very worst we're a 33 and a bit percent chance," Neeld said on Thursday.
"(But) I think at one of the other clubs, Carlton (coach) Mick (Malthouse) had a reasonable relationship with him as well.
"(Assistant coach) Leigh Brown and I rolled around to see Dawesy straight away and he let us in, so that was a good start.
"That was also part of the Mitch Clark (recruitment) 12 months ago, having Jade Rawlings on our coaching staff, who coached Mitch while he was in Brisbane. That's really important.
"We know what type of player Chris is and we'd really like to have him at the Melbourne footy club."
Neeld confirmed he met with Essendon forward Scott Gumbleton, who is weighing up a new offer from the Bombers and interest from Fremantle but gave no indication where his preference lies.
"I thought I was incognito over in Scarborough yesterday but that didn't quite work," Neeld said.
"Scott's in a position where he's weighing up his future. I just wanted to have a chat.
"He certainly, like Dawesy, plays a position that we need to fill at the club."
Asked whether he would wait for Dawes to make a decision before turning more attention to Gumbleton, Neeld said: "Not necessarily. That'd be alright I reckon - Clark, Gumbleton and Dawes. They'd all get a game.
"We're happy to have a look at everyone that may be available. We owe that to the core group of players that we've got at the moment that we're really confident about their futures. "
Neeld said the Demons have not opened discussions with any other midfielders after missing out on former Magpie Sharrod Wellingham, who was traded to the Eagles.
"We realise that we've got a couple of healthy draft picks still left. We haven't ruled anything in or out for the right type of player," he said.
Melbourne have first-round picks four and 20 in the national draft.