Attention at Old Trafford will soon switch towards the summer transfer window, with the Red Devils having already got one notable deal over the line.
Wales international Bale is among those said to figure prominently on the Old Trafford wish list, with the 29-year-old former Tottenham star enduring a testing time in Spain.
Cole admits a proven performer would be a welcome addition to the United ranks, but believes a top-four finish or European triumph would be required in order for Solskjaer to secure the Welshman.
The former Red Devils striker said of the Bale rumours: "I think Manchester United should be going for all top players of that calibre. That's what Manchester United is about.
"Do those players want to come to Manchester United? That's the question.
"If Manchester United miss out on Champions League football... top players want to play Champions League football.
"When you are playing at that elite level, it's not all about money. People think it's all about money, but you speak to players at elite level.
"Do you want to play Champions League football or do you want a few more quid? They will say Champions League, because they want to win it - it's a special competition.
"So if Manchester United miss out on Champions League football, it becomes difficult for them to attract players. It can be tough."
Cole believes a top-four finish should be the minimum requirement for United in this and any other season, with the club needing to ensure that their sights continue to be set higher than merely qualifying for elite continental competition.
A man who helped to deliver European glory as part of an historic Treble in 1999 added: "Manchester United should qualify for the top four season in, season out.
"It's not about competing for top four, it's about competing for the Premier League title. It is as simple as that.
"Sometimes we overlook that scenario when we think finishing in the top four is like winning a trophy.
"For Manchester United it shouldn't be like winning a trophy. To win a trophy, you have to finish top."