Solskjaer made a very public attempt to sign Haaland in January 2020 as the striker prepared to leave RB Salzburg. However, despite holding talks with Haaland's camp, Solskjaer suffered a snub as the striker accepted a rival offer from Borussia Dortmund.
It was a surprising blow for Solskjaer, who had worked with Haaland at Molde and given the Norway international his first team debut.
Despite the rejection, former United teammate Keane wants to see Solskjaer try again for the Dortmund star.
Keane was calling United's victory over Newcastle United for TV2 of Norway when asked which player he'd like to see Solskjaer sign.
"Only one? Wow ... Haaland would not be stupid. I think they have to get a forward. I do not know what will happen to (Edinson) Cavani's contract this summer," said the Irishman, quoted by VG.
"(Kylian) Mbappé is not bad, but if I have to choose only one player that Manchester United should pick up, then I choose Haaland. Yes, Haaland. We need a striker.
"Will he go to Spain? Does Manchester United fit into his priorities? He knows Ole Gunnar a little from before when they worked together. Maybe he can go? He will be a very good signing."
Keane was also asked about his ugly challenge on Haaland's father, Alfe-Inge. The tackle on the former Manchester City midfielder eventually saw Keane fined and banned. The former Republic of Ireland captain would later be found guilty for bringing the game into disrepute after an admission of his motivation for the tackle in his autobiography.
But he insisted: "I wanted him to be in pain, but I never had any intention of hurting him. For me, there is a big difference between giving someone pain and hurting them."
- updated February 24