The Norwegian has yet to win a medal during his time as United boss, although he has overseen runs to numerous semi-finals.
United got to the last four in the FA Cup, the League Cup, and the Europa League last season but lost to Chelsea, Manchester City, and Sevilla respectively.
Should United extend Solskjaer's contract?
It means United have not won a trophy since the 2016/17 season under Jose Mourinho, when they lifted both the League Cup and the Europa League.
United remain in European competition this season, playing Spanish side Granada in the quarter-finals of the Europa League, and Gillespie believes that Solskjaer will be under significant pressure to win silverware next season if he again fails to win the competition this term.
Speaking exclusively to TIF, Gillespie claimed that mere qualification for the Champions League was not something to celebrate, with silverware required sooner rather than later.
He said: “It looks like we’re going to qualify for the Champions League this year. Is that enough for Man United when you look at the past and what they’ve achieved?
“If they don’t win the Europa League, that’s another season without a trophy, you’re talking four years now.
“That’s a real barren spell for a club like Man United.
“You’d expect that, if they don’t, something has to be delivered (next season), without a shadow of a doubt.”