Burke is on loan from West Brom, but has impressed for the Scottish Premiership leaders.
He shone again as Neil Lennon’s side progressed into the Scottish Cup quarter-finals with a 2-0 win over Hibernian.
Burke did have to be substituted late on after being injured during the win, though the knock doesn’t appear too major.
And Celtic fans want the club to turn his deal into a permanent one when the transfer window opens again.
One said: “Oli burke is a must sign in the summer, would take him over any of the loans we have.”
While another added: “Celtic need to splash the cash and sign Burke at the end of his loan. What a player btw”
And a third wrote: “Hopefully the Celtic board will try and make an attempt to sign Oliver Burke. Already he is a big presence in the starting 11 and gives us another option to our style of play.”
One other commented: “Celtic need to do everything they can to keep Burke, he gives them a different dimension.”
With a fifth writing: “Celtic need to think big and keep Olly Burke – the boy is 1st class.”
The win over Hibs was the second with Lennon in charge after he replaced Brendan Rodgers in midweek.
Rodgers became Leicester manager with Lennon returning to Celtic Park.
And Burke believes Lennon is a good choice for the boss.
“I wouldn’t say I was disappointed that Brendan has gone,” Burke said.
“I still feel I can learn a lot from the new manager and I’m looking forward to working under him.
“I’d never met him before but he seems really good, really passionate about the club.
“He knows what it’s all about so he’s the right man for the job.”