Savage, who came through United's academy with Giggs and later played alongside him for Wales, believes the club's most decorated player has been harshly treated.
Louis van Gaal was sacked on Monday and is expected to be replaced by Jose Mourinho, with Giggs still considering his own Old Trafford future.
Press Association Sport understands the 42-year-old has been offered a role at United but looks set to miss out on the top job for a third time, having served as Van Gaal's assistant after spending a year as player-coach under David Moyes and taking temporary charge when the Scot departed
"I feel sorry for Giggsy," Savage said
"I believe when Fergie went they should have been brave and given him the job instead of Moyes and then they should have given it to him when Van Gaal got it
"So now he's been overlooked again, that's three times now. I think on one of those occasions he should have got it. He knows Manchester United off by heart.
"Look at Pep Guardiola at Barca, Zinedine Zidane at Real Madrid. I've got sympathy with Giggsy. I would have liked to have seen them give it to him."
Giggs has to decide whether to stay under Mourinho, gleaning tips from one of the most successful coaches in the game, or leave and gain more responsibility elsewhere.
"If I was Giggsy now I would look to start my career as a manager somewhere else," Savage said.
"If it's lower down fine, as low down as he wants to manage, but don't forget he's got terrific experience and what he's won as a player is second to none.
"He's been involved in too poor campaigns from two previous managers and maybe he's been tarnished a bit but it wasn't him making those decisions.
"If Giggsy can go out and make his own decisions now, do well somewhere, he can come back and the United job will be his."
After joining United's academy on his 14th birthday, Giggs went on to make a record 963 appearances for the club, winning 13 Premier League titles and the Champions League twice.
United have not finished higher than fourth since Sir Alex Ferguson retired and Savage says the club will be all the poorer without their former star.
"If he left it would be a shame for United and a shame for Mourinho," Savage added.
"He's someone who knows the football club better than anyone so of course he would be extremely valuable to Mourinho.
"He's been a magnificent player there but he has to do what is best for him now and I think what's best for him is to go."
Source: PA