Mulgrew and McNamara share three former clubs in Aberdeen, Celtic and Wolves, as well as Scotland.
And the current Parkhead player was grateful for McNamara's influence when he was at Molineux.
"I got on well with Jackie, he's a really nice guy and I'm sure he will be a decent manager at Dundee United," Mulgrew told the Celtic View.
"I played with him when he had just left Celtic so he had years of experience behind him and obviously that helped me.
"He's a great lad and I sat next to him in the changing room, he has good craic. He comes across as quiet but that doesn't mean he's like that in the changing room.
"He has all the attributes to be a good manager, you could see that when he was a player. He has some great knowledge of the game.
"He'd left Aberdeen by the time I'd signed but he was at Celtic when I was a youth player.
"Growing up a Celtic fan I obviously watched him through the years, he was a great player for the club and I think he can do just as well as a manager.
"He got off to a great start two weeks ago and he will be trying to build on that so we should be careful when they come to Celtic Park."
McNamara has led United to victories over Rangers and Hearts in his first two matches in charge, with his team hitting six goals in the process.
"United always produce a tough game, with a new manager in charge especially," he continued. "They'll be wanting to make their mark for him and do it by causing us problems so we'll have to be up for it.
"They're producing good, young, Scottish talent, and guys like Johnny Russell and Gary Mackay-Steven are players we'll have to watch."