It has been suggested Aguero would like to stay in England with Chelsea and United among the clubs who have been tipped to make a surprise free transfer swoop for the Argentine, 32.
United may have a need for a new frontman with Edinson Cavani out of contract in the summer and the Uruguayan yet to sign an extension to stay at Old Trafford.
But Borussia Dortmund’s Haaland, who the Bundesliga side want at least £127million (€150m) for if the Norwegian is to depart the Westfalenstadion this summer, is the top target for the Red Devils.
Barcelona and Real Madrid both held meetings with Haaland’s representatives, his father Alf-Inge and agent Mino Raiola, this week as the pursuit of the 20-year-old hots up.
There is long-term interest in Haaland from United and they were keen on the towering centre-forward when he played for Molde and also when he played for Red Bull Salzburg, before he opted to join Dortmund in January 2020.
Yet even if they miss out, Solskjaer has ruled out moving for veteran Aguero, insisting it does not sit right with him that players can swap between rival clubs, as Peter Schmeichel and Carlos Tevez have famously both done.
“We’d all like the squad sorted as soon as possible but I’m not the one in the trenches with the negotiations but so far we’ve done a really good job,” Solskjaer said.
“We had to wait for Bruno Fernandes but we knew he was the one and that worked out well.
On the possibility of signing Aguero, he added: “When I played for Man United, if a rival club tried to sign me, and I had gone… well where’s the loyalty?
“It’s not for me. I know we’re professionals, but when you play for Man United you don’t go to Man City.
“We’ve had examples of that and I didn’t really agree with it. I won’t name names but you all know who I mean. Enough said.”
It has not yet been confirmed when the summer transfer window will open and close although it must be between the dates June 9 and August 31.
United are also keen to strengthen other areas of their squad including at centre-back and at full-back.
Despite the pursuit of Haaland, United boast the second-best attack in the division – according to goals scored this season – with Marcus Rashford key to that.
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Rashford’s agency have gone as far to say it’s been two years since the United academy graduate played while fully fit.
But Solskjaer said: “I don’t have any concerns about Marcus’ general fitness at all. Every footballer carries a bit here and there. Very seldom do you say you’re 100 per cent fit.
“I’ve been impressed by his resilience and robustness. He wants to play. He’s got a character that even if he feels something, he doesn’t want to miss games because he loves Man United and football too much to dip out.
“He wants to give everything. That’s all credit to him but, of course, the little knocks he’s carrying we have got great staff to look after him. But it’s not been too big a concern or worry.
“There has never been a risk that we felt will damage him. It is just the way it is when you’re a professional footballer with a game every three or four days.
“It’s been relentless this season so we’ve not had the time. Sometimes you can send them away for a week to a sunny place and that will hopefully come back again so that we can look after the players.”