Welbeck has come through the ranks at Old Trafford and has scored 29 goals in 138 appearances, including 10 in 35 games this season.
However, recent speculation claimed that Welbeck was considering his future at United after playing on the wing for large parts of the season under former manager David Moyes.
The 23-year-old has been linked with Arsenal and Tottenham Hotspur, but interim boss Giggs does not envisage Welbeck moving on in the summer.
"We won't be selling him," said Giggs, who saw youngster James Wilson score twice against Hull City on his debut in midweek.
"We showed that the other night that bringing young players through is an important part of the club's culture and history.
"We wouldn't like to lose someone like Danny because he has done that, he has been here since eight or nine. So we want to keep him."
Giggs also confirmed that he has been consulted on possible departures, and potential summer signings by United chief executive Ed Woodward.
He added: "Ed has asked at different times, 'What do you think about this or what do you think about?'
"That is part of my job as I see the players day in, day out, and I've seen them perform in the last few days."
Louis van Gaal is the favourite to replace Moyes as United's permanent manager and Giggs believes the new boss would benefit from keeping the likes of club stalwarts Phil Neville, Paul Scholes and Nicky Butt on the coaching staff.
"It's a decision for the new manager and something he definitely has to consider," said Giggs.
"I would say there are more plus points than minus points with what the lads bring.
"They know the culture, the history and the way the club is run. All that can definitely be a plus."
Reports have suggested that Giggs will be offered the job of assistant to the new manager, but the 40-year-old is still weighing up his options regarding his future.
The Welshman said: "I honestly don't know what is going to happen next season. I haven't experienced it before and I will go away, chill out and make a decision.
"I'm lucky that I have a lot of options. It could be a sad time but also an exciting time as well.
"It's a question of whether still to play, whether to stay here, whether to go somewhere else, whether to chill out for a year and take a step back as I've been in football since I left school.
"But I could get pushed next week on telling them what I want to do."