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However, Madrid are not ruling out the possibility of trying to bring in both superstar forwards.
They want to analyse the numbers and terms involved in a deal for Haaland to decide whether to compete for his signing.
Sources close to Barcelona have told ESPN that the Norway forward is a priority for president Joan Laporta, but there are doubts at the Bernabeu as to whether Barca could afford a deal that, with transfer fee, commission and salary, could rise to over €350 million ($373m).
There is a feeling in both Spain and Germany that City are best positioned to sign Haaland. The Premier League club have been working on his signing since last August and could offer superior terms to those from his LaLiga suitors.
Pep Guardiola wants to bring in a No. 9 to complete his squad. City tried to strengthen their forward line with Harry Kane last summer, and have since received €55m ($60m) in transfer funds with the departure of Ferran Torres to Barcelona in January.
With that in mind, Real Madrid believe they are behind City as a possible destination for Haaland, but want to assess the situation before taking a stance on whether to pursue the 21-year-old or rule themselves out.
President Florentino Perez has an excellent relationship with Borussia Dortmund's executives. Last summer there was some contact between the clubs to discuss what fee would be required to sign Haaland.
Haaland's father, Alf-Inge, and agent, Mino Raiola, visited Real Madrid's training complex at Valdebebas last April for informal talks, after meeting with Barca president Laporta -- who is close to Raiola -- earlier in the day.
Real Madrid's ideal scenario would be to unite Mbappe and Haaland in the same team.
Coach Carlo Ancelotti refused to speculate about that possibility in a news conference on Friday, when he said: "I can't answer that question; they aren't Real Madrid players. I like to talk about the players I have."
There had been reports that Madrid would like Haaland to stay at Dortmund for one more season before signing him in 2023, but that possibility has been ruled out by the player's camp.
Haaland wants to leave Dortmund this June, and he has made it clear that he would like to make a decision in the next few weeks to avoid further drama.
Real Madrid believe that the situation will be resolved this month, hence their desire to speak to Dortmund in the coming days before taking a decision.
Barca hope to reinforce their defence and midfield with out-of-contract players on free transfers this summer so they can focus all their funds on Haaland.
Barca cannot match the financial muscle of City or Madrid but are looking at ways of making the Haaland transfer possible.
Sources have told ESPN that the club are now in talks with LaLiga to receive a cash injection from the CVC investment fund, after previously rejecting any involvement in the deal.