McVeigh was appointed Essendon coach on Monday, with the club opting for the Sydney Swans assistant ahead of former teammate Hird, Fremantle's Jaymie Graham and Adelaide's Murray Davis.
Club legend Hird, who quit as Essendon coach in 2015 after a tenure marred by the club's supplements saga, had pitched for the top job.
Hird was an assistant to McVeigh when he was interim GWS coach in 2022.
"I love James Hird, he's one of my best mates," McVeigh said.
"It was difficult to go through the process against a good friend."
Amid speculation of a backlash from Hird's supporters, McVeigh added: "I don't fear those people. I actually embrace how they feel. And I understand it to be fair.
"He's been a constant in my life ... we spoke this morning. It hasn't broken us, we're still together. I embrace the feelings of people that love James Hird because I do too."
McVeigh speaks regularly with Hird and will discuss with him the prospect of a role at Essendon.
"We'll have those discussions going forward, and it'll ultimately be up to James what he wants to do, and I'll respect that," he said.
Pressed on whether he would like Hird back, McVeigh said: "Once I found out I got the role, he was certainly at the top of my list, there's no doubt about that."
Hird didn't answer Essendon president Andrew Welsh's calls on Monday.
Welsh texted Hird and hoped to speak with his former teammate on Tuesday.
"I would love to see James as part of the club going forward. He's such a fantastic football person," Welsh said.
"I feel there is significant value James can bring. Not just tokenistic, but real value James can bring to our club and playing group."
Murmurings have already arisen of a board spill by Essendon fans disgruntled Hird was overlooked.
"I sit here year after year and we get people coming out of the shadows that want to challenge the board," Welsh said.
"I can tell you, since I've been here, this club has never been more united.
"So these people want to keep living in the shadows and come and try and destabilise our club. We don't jump at shadows.
"We're a strong club, and we'll continue to make sure that we're making the right decisions on behalf of our members and our teams and our staff to get this club moving forward again."
McVeigh "absolutely" wants ex-caretaker coach Dean Solomon, who was on hand at the announcement, on his coaching team.
McVeigh was one of the banned "Essendon 34" and couldn't fulfil his role as an assistant at the Giants during his suspension.
The supplements saga has hung over the club, but their new coach was adamant he could drive Essendon past it.
"That was a long time ago," McVeigh said.
"Tested my resilience, so it helps. But we won't live in the past. We're here for the now."
McVeigh, who played 232 games for Essendon, described himself as a "no ego" coach as he bids to lift the Bombers from the abyss after they collected the wooden spoon in 2026.
"I want to bring the club back to where it needs to be," he said.
McVeigh has only briefly met champion midfielder Zach Merrett, who is expected to seek a trade to Hawthorn for a second straight year.
"You must respect your champions. I'm big on that," McVeigh said.
"Zach and I will sit down at the appropriate time.
"I think it's really important we understand the player's feelings. It's not just about the club's feelings, it's about his feelings and where he's at.
"I respect him. I want him to be here, but I'll listen to him as well."
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