The midfielder referred to moving to the Bernabeu as a dream during an interview when he was on international duty with France in March.
Pogba was desperate to leave the club for Barcelona last summer after falling out with Jose Mourinho but Solskjaer has reinvigorated the midfielder since taking over from the Portuguese coach.
Nevertheless, Pogba is believed to be keen on working with Zinedine Zidane and his recent comments did not go down well at Old Trafford.
Though he dismissed it publicly, Solskjaer pulled Pogba aside at Uniteds training ground last week and asked to see the midfielder in his office.
According to the Mirror, the Norwegian made it clear to Pogba that he would not be sold this summer and told him to forget about Real Madrid.
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Pogba has over two years to run on his existing deal at Old Trafford and United have the option of extending that by a further 12 months.
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