Giggs was handed the reins for the last four games of the season when David Moyes was sacked this week, but he is not thought to be under consideration beyond that.
The 40-year-old Welshman, a one-club man who has won every major honour at Old Trafford, has long been touted as a future United manager but, with his playing career still ongoing, the job may have come too early on this occasion.
But Bruce, who played alongside Giggs during his time as club captain, believes he could yet edge out high-profile bosses such as Louis van Gaal and Carlo Ancelotti.
"There's been a lot of players who have jumped in - (Pep) Guardiola springs to mind, (Jurgen) Klinsmann springs to mind - big players in big jobs. And the one thing it won't do is faze Ryan, he's been at Man Utd all his life," said the Hull manager.
"If they win the next three or four games 4-0 there'll be a clamour for Ryan I'm sure and you wish him the best of luck.
"He has nothing to lose and no pressure on him, the supporters will be right behind him. He might win the next four games and give them something to think about.
"Make no mistake when you do your first team talk or pick your first team it's hard and you understand what a big job it is.
"But the one thing that will never daunt him is the Man United thing, because he's been there all his life.
"He's got a group around him who all know the Man United way and I'm sure Old Trafford will be behind him."
Bruce was also keen to offer a word of consolation for the axed Moyes.
News of the Scot's sacking was broken in the media on Monday afternoon before Moyes was officially told on Tuesday morning, and Bruce was disappointed to see the decision leak from his former club.
"Looking from the outside you'd have to say, and I'm sure Man United feel the same, that they could have handled it better," he said.
"To find out on social media...we know the way the world works now, but I'm sure nobody would like to lose their job like that.
"The one thing you do, and I'm sure Dave is doing it now, is take it personally.
"I'm sure he'll have a little break, reflect on it and come back better. But it's certainly been a difficult few days for him."
Bruce was once viewed as a potential United manager himself, given his connections with the club, but while he would clearly relish such an opportunity he admits it something of a long shot.
Asked if he would had been eyeing the vacancy, Bruce laughed.
"Silly question," he said.
"I've done 10 or 11 years in the Premier League and what's the next step? That's the question I keep asking myself, for that we'll just have to wait and see.
"You seem to get pigeon-holed but you never give up. You never stop wanting to get to the top.
"What they're looking for is people who've won trophies and no disrespect to the club I work for now or Birmingham or Sunderland, but you can't win the Premier League with those clubs.
Like Bruce, Ole Gunnar Solskjaer believes his former team-mate Giggs ticks all the boxes to become a successful manager.
Cardiff boss Solskjaer has spoken to Giggs this week, and he will also be looking for United to do the Welsh club a favour with their next two games being against Cardiff's fellow relegation candidates Norwich and Sunderland.
"I have spoken to Giggsy, of course, and I wished him all the best," Solskjaer said.
"A boy from Cardiff should do us favours now. The next two games are important for us.
"Giggsy has got all the attributes to become a top, top manager one day, definitely.
"When he was a player - he still is a player - he chooses his words very carefully.
"He is not one of those that always speaks every single day, but when he speaks you listen. It has always been that way. He commands respect."
Solskjaer also revealed that he had spoken to Moyes, who, it appears, found out about his pending dismissal earlier this week through leaks in the media.
"I don't think it is for me to talk about," added Solskjaer, when asked about Moyes' exit.
"You are sorry to see managers lose their jobs. I have had contact with David and thanked him for his help, because we brought Fabio and he has helped us with Wilf (Zaha).
"Manchester United will always be challenging for the top places, so they will be fine. They will make the right decision (on a new manager).
"Stuff leaks in the media all the time. I don't know what happened, so I can't comment on it."
Arsenal manager Arsene Wenger, now far and away the longest-serving Premier League manager having taken charge of the north London club in 1996, was dismayed that United had not stuck with Moyes and warned the high turnover of coaches would affect the quality of coaching in the long term.
"If you want quality people in any job, you need to give them time to develop and to become good, or people with the quality will not come into our job any more," he said.
"The average (job life) expectancy of an English professional club at the moment is 11 months, and that is quite unstable. Every guy who is married, has a family, will have a big hesitancy before he goes into that game.
"That means the quality of the coaching and the quality of the managing is under threat."
Crystal Palace manager Tony Pulis believes Manchester United's reputation has been tarnished by the departure of Moyes after just 10 months in charge.
The former Stoke boss believes Moyes deserved more time to make an impact at Old Trafford and feels the club's image has been damaged by the manner of his exit.
"Personally I'm disappointed, really disappointed for David. I think he deserved longer and the way it was handled was disappointing as well," Pulis said.
"Manchester United for the past 26 years have been a beacon of how to run a football club in respect of stability and how the club was built and how it was run.
"A lot of that has been washed away over the last few days in how they handled the situation. That's my personal view."
Manchester City boss Manuel Pellegrini said it was "a pity" that Moyes had been sacked so early into his United career.
Pellegrini said: "It is a pity for David Moyes because I think he is a very good manager, that is why Manchester United chose him to continue the work that Sir Alex Ferguson did for so many years. But I don't know the reasons they have. It is always not good for the club to sack a manager who has a contract who is just starting his work - and very difficult work after Alex Ferguson and a contract for six years."