United return to the competition on Tuesday night with a clash against Swiss outfit FC Basel having competed in and won the Europa League last season.
Giggs believes Jose Mourinho’s side are a ‘little bit short’ to win the Champions League this term but says the arrival of a player like Bale would change their prospects.
The Wales legend 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.’
Giggs, who became the most decorated footballer in history during his 14-year spell at Old Trafford, added: ‘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.’
Bale has developed into one of the best players in the world over the past five years, scoring 55 goals in 103 league games for Real since leaving Tottenham in 2013.
The Wales international was linked with a move to United during the summer window but says he stayed in the Spanish capital to win trophies.