Parachuted in for the rest of the season when Jose Mourinho was sacked in December, the 1999 treble hero has presided over a remarkable upturn in fortunes.
Solskjaer has lost just one of 17 matches since taking over and managed to render that 2-0 loss to Paris St Germain meaningless by overseeing an extraordinary last-16 second leg win in France.
Marcus Rashford's last-gasp spot-kick sealed a 3-1 win that saw United make history and progress to the quarter-finals on away goals after the tie ended 3-3 on aggregate.
Lukaku's first-half brace gave United hope and the striker fully expects the Norwegian to get the job.
"I know he's going to stay, there's no doubt about that," he told Viasport. "He wants to stay, the players want him to stay. We're doing really well, we're playing like Manchester United should play.
"He's a young coach, he has young players as well so it's the perfect environment to develop and hopefully win trophies in the future."
Lukaku pointed to the wins at Tottenham, Chelsea and Arsenal in another interview with beIN SPORTS, before asking: "What else does he have to do?"
United are into the Champions League quarter-finals for the first time since 2014 and have broken into the top four, having been 11 points off it when Mourinho was sacked.
Asked what Solskjaer has done, Lukaku said: "He's done a lot. When I came back in, for me, personally he had a plan.
"He wanted me to be really sharp in three or four weeks and that's what happened. We took three or four weeks, I did a lot of interval work and a lot of finishing and now I can say that I am fully fit.
"But then for the team, he made us play offensive football. He gives the young players confidence. Sometimes he's tough but at the right time he knows also to give a compliment.
"In training we do a lot of small games, a lot of focus on our offensive part of the game and that's what the players like. So, we just try to come in every game and win the games."