Grimsby, who are currently sitting 21st in the fourth-tier of English football, parted company with previous boss Paul Hurst last month and Healy emerged as the favourite to fill the managerial vacancy at Blundell Park in recent days.
Football Insider reporter Pete O’Rourke tweeted on Monday evening that Healy ‘is in the frame’ for the role and BetVictor have since slashed his odds from 6/1 to 2/1 and now sit at 1/2 as of Tuesday afternoon.
Healy has enjoyed a hugely successful eight years in charge of Linfield, winning the Premiership title on five occasions and his side have already opened up an eight-point gap at the top in this campaign.
He has also led the Blues to two Irish Cup crowns and two BetMcLean League Cups, including last season’s final triumph over Coleraine at Windsor Park, while winning 266 of 420 matches across all competitions.
After moving to Manchester United’s academy in 1995, Healy enjoyed spells in England with Port Vale, Preston North End, Norwich City, Leeds United, Sunderland, Ipswich Town, Doncaster Rovers and Bury as a player.
"I was surprised to be honest,” he told UTV Sport ahead of tonight’s BetMcLean League Cup clash with Glentoran. “I've always said and will continue to say that I've a big job here at Linfield.
"It's a privileged role I'm in. I've been loyal for the eight years I've been here in terms of the hard work and honesty and hopefully success I've brought here, so it came as a surprise to me.
"I got a few messages last night and more this morning from people reading into whatever it is.
"My concentration solely is on managing Linfield with Glentoran at The Oval tonight and Larne on Saturday."
The 44-year-old has been linked with numerous jobs across the water in the past and previously stated his desire to move up the managerial ladder, but says the timing has to be correct.
"The timing has to be right of course,” he added. “I'm under contract for another year after this one.
"Eventually do I want to manage at a bigger level or better level possibly across the water? Of course, everybody does.
"In any job role you go into you want to improve, get better and get to the top of the business you go into.
"I'm privileged and I love managing this football club. It means a great deal to me.
"That's why at the minute my sole concentration and full focus is on Glentoran tonight and Larne on Saturday."