In-form Hooper took his tally for the season to 16 goals with an excellent touch and lob in the 2-0 Scottish Premier League win over Dundee on Boxing Day after starting the game just off the front two.
The former Scunthorpe striker has been tipped for an England call-up following his Champions League exploits for Neil Lennon's side, and Mulgrew is eager for Hooper to stay beyond the January transfer window.
Mulgrew said: "Hoops can play that role (off the front two). He has played there a few times but he will always get goals, no matter where he is on the pitch.
"He is good at holding the ball up and he's a great finisher. Even when he's a striker he comes off - you can see the interchange.
"He is playing really well. He's a good player and he's coming into his prime. As long as he keeps playing well it will be good for us.
"We want to keep the squad together and I'm sure the manager does as well, so hopefully there are no bids for him.
"Every player wants to play in the Champions League, especially the last 16."
Scotland international Mulgrew started at centre-back against Dundee but went off at half-time with a rib injury and is a doubt for Saturday's clash against Hibernian at Easter Road.
Celtic are looking to maintain or extend their seven-point lead at the top following their fifth consecutive SPL victory.
Mulgrew said: "It was a good result and another three points. It wasn't pretty but they are the games you have to win.
"We don't want to get ahead of ourselves. It's only five wins and we have to focus on the next game against Hibs."