The Red Devils legend feels the current squad, when taking into account possible outgoings, is “four or five” bodies short of being where it needs to be.
United will be under pressure to mount a Premier League title challenge next season, having spent much of the current campaign focusing on cup competitions.
Big-money additions are expected to aid that cause, with Giggs of the opinion that Mourinho will have gaps to fill all over the field.
He said in The Express : “He needs to replace Michael Carrick.
“[Zlatan] Ibrahimovic is going to be out for a year. I think he needs a centre forward.
“Don’t know what’s going to happen with Wayne [Rooney].
“I think he probably needs four or five players.”
With uncertainty shrouding the respective futures of frontmen Ibrahimovic and Rooney, United have been linked with several potential alternatives.
The likes of Antoine Griezmann, Andrea Belotti and Romelu Lukaku have generated the most speculation, but Giggs feels Mourinho should be looking to raid a top-four rival.
With Sanchez about to enter the final 12 months of his current contract at Arsenal, the Chilean is considered to boast the qualities that could make a big impact at Old Trafford.
Giggs added: "Jose Mourinho bought [Zlatan] Ibrahimovic because he is a winner.
"He'd won everywhere he'd gone. Not only the effect on the pitch but off the pitch as well.
"You can just feel that Sanchez would give his team-mates a lift."
Sanchez has netted 26 times in all competitions for Arsenal this season, with Ibrahimovic the only man to have bettered that return at Old Trafford.