Rashford came through the United academy while the club was still winning trophies with Sir Alex Ferguson in charge, but the Red Devils have not won the Premier League since the legendary manager retired in 2013.
Jose Mourinho was manager when the club last lifted silverware back in 2017 with League Cup and Europa League glory, while current boss Solskjaer got to three semi-finals last season but lost them all.
United have already been knocked out of this season’s Champions League at the group stages, but are still competing on four fronts with the Europa League, Carabao Cup, FA Cup and Premier League now their aims.
But to compete with champions Liverpool – who currently lead the top flight – Rashford says the team need a little extra to unlock their potential.
“I feel like we are close, but we are not quite there,” he told the Official Manchester United Podcast.
“We just need to make that final push to be a team that can win trophies and fight for everything.
“In the big games, we go toe-to-toe with any team and we know that we can do it.
“We have confidence on the pitch that we can beat them, whether it is Liverpool, (Manchester) City, PSG (Paris Saint Germain) against the big teams we show that we can go head-to-head and give them a game.
“The players know that, the manager knows that. We are just so close to that bit when you are winning trophies, but for me the sooner we get to that, the better.
“There is nothing bigger than winning the Premier League or the Champions League – that is what the dream is.”