It comes after the revelation from Bombers board member Kevin Sheedy that he voted for Hird to return as coach, which was at odds with the club's claim that Brad Scott's appointment was "fully endorsed".
Barham said Hird went through the "exact same process" as other coaching candidates, including Melbourne assistant Adam Yze.
But former North Melbourne coach and AFL football operations boss Scott was ultimately deemed the best fit.
"Hirdy will always be a champion of this football club," Barham said on Friday.
"I can't speak for him on whether he wants to get involved or whatever, that will be something for him, but he'll always be loved by this footy club."
Sheedy told the Herald Sun he was disappointed in the Bombers' media department for releasing a statement that said the board was united in appointing Scott.
The four-time Bombers premiership coach revealed the board's vote was 6-1 in favour of Scott, and declared he felt Hird deserved a second chance after stepping down as coach late in the 2015 season.
But Sheedy added he would now support Scott, who was officially confirmed as coach on Thursday night.
The appointment came after Essendon parted ways with Ben Rutten in August and failed to land primary coaching target Alastair Clarkson.
Four-time Hawthorn premiership coach Clarkson instead opted to join North Melbourne.
But he has delayed his start at the Kangaroos because of an AFL investigation into racism allegations from his time at the Hawks, which he denies.
Barham has "occasionally" reflected on the sliding doors moment when Essendon missed out on Clarkson.
"I feel for everybody in that (North Melbourne) process," Barham said.
"I've seen (Kangaroos president) Sonja Hood at a couple of things and I feel for her with the position she's in, but I have no regrets with what we did."
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