Robinson has been isolating at home ever since coming into contact with a Republic of Ireland team mate with coronavirus whilst on international duty.
The striker has since tested negative for COVID-19. But government rules state he still needs to isolate for 14 days.
And Albion are still waiting to hear whether he has to miss the Burnley game or whether he will be able to play if he continues to test negative.
“It’s not in our hands,” Bilic said.
“He was with Ireland. One of the players who was positive, Callum was one of three or four players playing cards with him or sitting next to him on a plane.
“The rules are different in Ireland than they are in England.
“He has had to self-isolate – 10 days, two weeks? It’s a bit complicated to be fair.
“We have to hope that he will be available for training on Sunday.
“That wouldn’t put him in the greatest condition but he has played a lot.
“He played four games for us, he played one game for Ireland.
“We will be happy if he comes back, we will take that option.
“But is it going to happen? We don’t know. It’s not in our hands. We are going to see what the government says.”
While Robinson remains a doubt, Albion will definitely be without fellow striker Hal Robson-Kano who broke his arm at Southampton.
And Bilic revealed he expects the Welshman to be out for around a month.
“He had surgery and it all went really well," the Croatian said.
“It’s going to be four or five weeks.
“The main thing is for the stitches to heal – after that it will go very quick.
“But they have estimated, the doctors, four or five weeks.”
Meanwhile, Albion saw Kenneth Zohore join Millwall on loan on deadline day.
Kamil Grosicki also agreed a loan switch to Nottingham Forest.
But it has not yet been confirmed if the paperwork was submitted on time.