The Red Devils made three key signings before the transfer window closed on Thursday, spending £140 million on Harry Maguire, Aaron Wan-Bissaka and Daniel James.
Despite losing Ander Herrera to Paris Saint-Germain earlier this summer and Marouane Fellaini in February, the Red Devils did not make any additions in central midfield.
Wales manager Giggs, who is United's record appearance-maker, said he has concerns about the club's engine room ahead of their Premier League season opener against Chelsea at Old Trafford on Sunday, per Rob Dawson of ESPN FC:
"I think we need a couple more midfielders. If Paul Pogba goes—we talk about culture change—but, obviously, you need quality. You might not like it, but come Sunday if Paul Pogba isn't in that midfield or the week after, and we've got the midfielders that we've got at our club, then we're short. That's the problem."
Pogba, 26, will be crucial for manager Ole Gunnar Solskjaer in 2019-20 if United are to break back into the Premier League's top four.
The Frenchman has blown hot and cold throughout the three seasons since he returned to the club from Juventus.
His best period was the 10 weeks in the immediate wake of Solskjaer replacing Jose Mourinho back in December, when Pogba played a key role in United winning 14 of 17 matches in all competitions.
Scott McTominay is likely to partner Pogba in the centre of midfield in the new campaign, and the 22-year-old has impressed of late:
Other options at Solskjaer's disposal in the midfield include Nemanja Matic, Andreas Pereira and Fred.
All are talented but inconsistent, and if one or two of them get injured, United could find themselves lacking options, especially when the games come thick and fast around Christmas time.
United likely have plans to strengthen their midfield in the future, but they may pay for not replacing Herrera and Fellaini this summer.