Following last year’s poor campaign, Bruce is eager to freshen up his squad this summer with five or six new additions.
Palmer, meanwhile, is set to compete with David Button for the number one jersey after putting pen to paper a new four-year deal at The Hawthorns.
The keeper made just one first-team appearance last season.
But he has previously enjoyed successful loan spells with Plymouth and Lincoln.
“Steve Bruce is a man with great pedigree who has won promotion with a lot of the clubs he has been at,” said Robinson, who played more than 200 games for the Baggies after arriving from Watford in 2003.
“The board have got to back him. They have got to give him what he wants. He has to be able to bring in players that suit his style and how he wants to play.
“That is why you bring these managers in – and you have to give them time.
“It’s only after they’ve had that time that you can judge whether they are a success. We have to see what happens, the players that come, the players that go.
“I saw Alex Palmer signed a new four-year deal. I really like him as a goalkeeper I think he has got great potential. He did fantastically well at Lincoln. You have to give those players an opportunity to prove themselves.”
Meanwhile, Albion centre-back Cedric Kipre has again been linked with a move to Scottish giants Rangers.
Kipre has struggled for minutes ever since arriving at The Hawthorns in a 900k deal from Wigan in 2020.
The centre-back made just one substitute appearance under Brice last season.
And with the boss eager to cut his wage bill so he can overhaul his squad, it is understood Albion will now consider loan and permanent offers for the 25-year-old.