The Norwegian still wants to bring in another two high-profile signings before the window closes on October 5, with Borussia Dortmund winger Jadon Sancho remaining his priority target.
Ahead of United’s season opener against Crystal Palace on Saturday, Solskjaer admitted he understood supporters’ frustrations over a lack of additions, with Donny van de Beek the only arrival so far.
Meanwhile, rivals Chelsea and Manchester City have been active in the market in a bid to close the gap on champions Liverpool, who completed the signing of Champions League winner Thiago Alcantara on Friday.
Chelsea, who finished behind United on goal difference, have brought in a series of marquee signings in Timo Werner, Kai Havertz, Hakim Ziyech, Thiago Silva and Ben Chilwell.
Despite Frank Lampard’s high-profile additions, Solskjaer claims the limited number of players capable of improving his United squad has had an impact on his ambitions this summer.
“Of course I want players in the building that we want to work with and who want to be a part of it and who want to be here,” he said.
“One, you need the quality on the pitch. You also need the personal qualities and to really want to be here, and you need to show that you want to be here.
“That’s really important for us as a group, that we move forward.
“It’s not as big a pool as for other clubs, because part of Man United and being good enough is one thing. Then you’ve got another club that probably want to keep that type of player of that quality.”
The Covid-19 pandemic has also forced United to redraw their plans after executive vice chairman Ed Woodward had initially intended to heavily back the latest stage of Solskjaer’s Old Trafford overhaul.
United are refusing to meet Dortmund’s £108million valuation of Sancho, with talks now at an impasse.
The club’s hierarchy are reluctant to go above £70m on any individual target, prompting them to consider short-term options such as Ivan Perisic, Douglas Costa and Gareth Bale before the Real Madrid winger opted for a move to Tottenham.
“You can’t ignore the impact on the coronavirus,” said Solskjaer. “I think that’s really changed the market. It has been more or less in a period that maybe negotiations will happen.
“Of course the financial uncertainty of that has added to the difficulty of predicting where we’re at. You want always to have as long as possible with players.
“I think last season though shows that good players can adapt and get into a group quickly. Harry [Maguire] came in very late in the transfer window and had a great impact on the performances in the team.”
City have also began to respond to be dethroned by Liverpool with the signings of Nathan Ake and Ferran Torres, while Pep Guardiola is also keen to bring in Kalidou Koulibaly from Napoli.
Solskjaer added: “My focus is us, of course. You see other clubs and what they do, they put a challenge up for us.
“Our response has to be results. That’s the response. We can be best at the things that don’t cost anything.
“The culture here the training the coaching, getting fitter. But yeah, we’re looking at a small pool of players that can add to the quality of our team, and let’s see what we can do. They are working hard and I understand that they are working hard.
“We’ve done some business with Donny. We are very happy and delighted to get him in. He’s a very good player.
“I can’t say about other clubs’ players. I can’t speculate. We have to develop what we have and strengthen the team. Let’s wait and see if we manage to get someone in. I’ve been in dialogue with the club, we are always discussing, let’s see if we can bring anyone in.
“Fans will always want the best possible team to watch - the best Man United team,” he added. “That’s the passion we want.
“Over our history we can look back 10-20 years and you look back at the great players we’ve had. Going forward, since Sir Alex Ferguson left, we’ve averaged fifth place. Last season we were third and it was a big step.
“Everyone saw the development of the team, Bruno (Fernandes) coming in, Harry as captain with his leadership skills. Aaron (Wan-Bissaka) did really well. We want to step forward, develop and go into season with confidence, looking to give the group a boost. We are working on it.
“I think we can go into new season with confidence with the way we finished last season. That showed what we are capable of. We developed throughout the season and will keep on developing. We’ve shown our potential, but need more consistency. With a year more experience, we should get consistency.
“I don’t want to say what they cannot do. They can go as far as they want themselves.”