Solskjaer guided United to three semi-finals last season but could not reach the final in any of the FA Cup, League Cup or Europa League.
They are in another semi on Wednesday, taking on rivals Manchester City for a spot at Wembley in this season’s League Cup final.
Solskjaer has previously mentioned how “delighted” he is to reach semi-finals, though Keane believes sights need to be set higher.
The former midfielder said on Sky Sports: "Well you need to start somewhere and I think this is the trophy that realistically they have their best chance of this season.
“A one-off semi-final against your local rivals at home, if they can get a win, continue their decent home form which was really poor at the start of the season.
“I think will be massive. A really important game for Man United tonight.
"And for Ole. I know he's mentioned before about getting to semi-finals. Getting to semi-finals isn't good enough for Manchester United. They have to get over the line.
“Tottenham waiting, Mourinho. Nice ring to it. Fingers crossed they can do it."
Solskjaer is hoping his side can go one better this year having been knocked out by City in last year’s League Cup.
“It’s a chance to get to a final and, for this team, it would be a very, very big step, getting your hands on a trophy.
“I think we’ve improved immensely in a year since the last semi [against City]. We want to go one step further. I speak from my own experience.
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“When I won my first league [title] at Man United, it becomes a drug, celebrating with your team-mates on the pitch with a trophy.
“It becomes a thing, you become addicted to winning and that win [in the League Cup in 2006] with Patrice [Evra] and Nemanja [Vidic], I think it was huge for them and that team. I was on my way out then but they have spoken about it.
“That team grew so much from those years. We have added some players in our group that hopefully can give us that extra edge in the semis compared to last season.”