The former West Bromwich Albion midfielder previously advocated the appointment of his former captain Nigel Pearson.
Moore has taken four points from his two games in charge, overseeing a 1-1 draw against Swansea City and a shock 1-0 win against Manchester United.
And Palmer, who previously advocated the appointment of his former captain Nigel Pearson as Alan Pardew's permanent successor, hopes the ex-Albion defender will benefit from the Rooney Rule - which requires clubs to interview at least one BAME (black, Asian and minority ethnic) candidate - as long as he continues to get results.
"Now, we have all this talk about the Rooney Rule, let’s see it in effect here if Darren Moore does a good job between now and the end of the season," the former West Brom midfielder told Proper Sport on Monday.
"I was told by my pals who are big West Brom supporters that the spirit seemed to have returned, because Darren’s that type of bloke. He wasn’t a technically gifted lad but he was 100 per cent and (he has a) camaraderie with the team so I can see him bringing that to the table, and if they get results from now until the end of the season then that Rooney Rule should come into effect and they should appoint him.
"He’s an upstanding character, he’s done all his coaching badges, so why not? They put Alan Pardew in charge who’s had job after job after job when he’s failed. So why not give somebody a chance - a black guy a chance?
"They’re going to be in the Championship, so why not give him the opportunity - rather than bringing in a foreign manager and paying him a lot of money, or bringing in another manager that’s managed in the Premier League or the Championship and has failed?"
Despite Moore's early success, West Brom have only four games left in which to overturn the nine-point deficit between themselves and Swansea City.
Carlos Carvalhal's Swans do however have a game in hand on bottom two sides Stoke City and the Baggies.
Should Darren Moore come under West Bromwich Albion's consideration?