Bombers legend Hird had been vocal in expressing his desire to return for another stint as coach, after Brad Scott was sacked in May.
Fremantle assistant Jaymie Graham and Adelaide assistant Murray Davis were also in the mix.
But McVeigh, an assistant to Dean Cox at Sydney, is reportedly on the verge of being appointed, with an announcement expected as soon as Tuesday.
Interim coach Dean Solomon hands over the reins after the Bombers (2-21) completed the worst season in club history with a 10-point defeat to Port Adelaide on Sunday.
"Firstly, congratulations to Mark. He's a great friend of mine and I'm very proud of him that he's going to be named the coach, so good on him," Hird said on Nine on Sunday night.
"From a personal point of view, it's very disappointing. I'm disappointed. I put my heart and soul into the effort, but obviously fell short and that's the way it goes."
Hird indicated he was yet to hear an official decision from Essendon, but said "maybe that's coming".
The 53-year-old ruled out returning to the Bombers in a different role, but said he still has the passion to coach.
Hird is director of coaching with VFL club Port Melbourne.
"We have a finals series to play," Hird said.
"We'll see what plays out at the end of the year with some other clubs who have contacted me and we'll see what happens."
McVeigh is reportedly set to present to the Essendon board again on Monday, with an announcement imminent.
The 45-year-old played 232 games for Essendon and was one of the 34 players suspended over the club's devastating drugs saga.
McVeigh currently serves as Sydney's defensive coach, but could leave the club immediately to play a role in the Bombers' end-of-season exit meetings and list management decisions.
The second-placed Swans are preparing for a finals campaign while bracing for more fallout from the sexual assault allegations that have rocked the club.
"That's one thing this football club's done well for a long period of time ... be ready for the unexpected or if something does arise," Sydney coach Cox said last week about the potential of losing McVeigh to the Bombers.
"We've had those meetings (and) discussions. We're really clear on what will happen if he was or wasn't to get the (Essendon) position."
Two-time Essendon premiership champion and former captain Hird first took over as Bombers coach in 2010.
He was suspended for the 2014 season amid the fallout from the drugs saga, returned as coach in 2015 and resigned in August that year.
Hird had garnered plenty of support from Bombers fans in his latest bid to return as coach, after he narrowly missed out when Scott was appointed in 2022.
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