"We let Alexis and Romelu go, you don't have to be a rocket scientist to see that it's a forward that next time we're going to recruit. We're looking for some creativity and goals," Solskjaer said.
A move for Juventus forward Paulo Dybala collapsed in the final week of the transfer window last month and there were tentative inquiries about his club teammate, Mario Mandzukic. The departure of Lukaku in a £74 million move to Inter and Sanchez on loan to the Italian club has left United light up front.
The situation has been compounded by injuries to Rashford and Martial and placed added onus on 17-year-old Mason Greenwood, who has two goals in three starts. Rashford is not certain to play before the international break due to a groin injury. Martial is being given every chance to prove his fitness for Arsenal after five weeks out with a thigh problem.
"If the right one [had been available], if it was there, we would have [signed a forward]," Solskjaer said. "It was time for Rom to go. He knows that. He didn't want to be here. What is the point in having players that don't want to be here?
"And there's no point getting players in that you're not 100 per cent sure about. When you get players in, you need the right ones that are going to stay here for a longer period and that's the long-term thinking we have to show. I cannot think, 'I need a player because this is my reputation'. No, it's the club.
"There were strikers [available], but they weren't the ones we wanted. We couldn't get the one we wanted."
United are 11th in the Premier League and only Crystal Palace among the teams ahead of them have scored fewer goals than Solskjaer's side. Rashford and Daniel James are the top scorers with three goals apiece. Solskjaer admitted his team need more cutting edge.
"We need to score more goals, definitely, and that's what I've said here," he said. "We're working on improving relationships, the patterns [of play], the understanding and we've started in one end [of the pitch] with the foundation. We've done great defensively but we're not happy with the return going forward."
United's troubles this season have also been exacerbated by an injury to Paul Pogba, whom Solskjaer is hoping will be fit to face Arsenal after the France midfielder suffered a recurrence of an ankle injury against Rochdale last Thursday.
The over reliance on Pogba for goals and creativity from midfield has become a pressing problem. None of United's midfielders have scored this season.
"You wouldn't mind scoring a few goals on set-plays and from midfield, of course," he said. "The forwards are always going to give us goals so, of course, yeah, we both expect and hope they're going to come."
Solskjaer said Manchester City and Liverpool were beyond reach for now but believes United should be competing for one of the two top-four berths behind the title chasers.
"We should be in and around the places behind there. Chelsea are going through a similar period of rebuilding, Arsenal, Tottenham. You have got Leicester, there are loads of teams there wanting those positions and that's where we'll be, in and around."
