The Red Devils are back among Europe’s elite for 2017-18, with a Europa League triumph in Jose Mourinho’s first campaign at the helm offering them a route back to the top.
They will open their campaign with a home date against FC Basel on Tuesday and are being tipped by many to enjoy a positive return to the continent’s premier club competition.
Giggs believes United are heading in the right direction, having spent big once again over the summer on the likes of Romelu Lukaku and Nemanja Matic, but fears they may come up short against the likes of Real Madrid, Paris Saint-Germain and the very best in the business.
The two-time Champions League winner told Sky Sports: “You look at the current Manchester United team and you wonder if they can they win the Champions League? Have they got the players to win it?
“They're probably a little bit short at the moment. You've got Paul Pogba and Romelu Lukaku, if he fulfils his potential, Marcus Rashford, David de Gea, a world-class player, and then if you add somebody like Gareth Bale to the mix, that's when you talk about United having Champions League expectations.
“Obviously, Mourinho has won the trophy before so we're going to be there or thereabouts, but whether we've got the players to go the whole way, I don't know.
“Apart from Real Madrid, obviously you've got PSG, who have made big intentions this year, but it could be quite open otherwise with Barcelona losing Neymar and not making the best start.
“You have to look at Real Madrid, what they've done after the last two years and the players they've got, as the team to beat.”
Madrid remain the team to beat in Champions League battle, having made history by claiming the crown in back-to-back campaigns.
United, though, have made an unbeaten start to the Premier League season and will be in buoyant mood regarding their chances of going deep into the competition.