His agent Mino Raiola is working away on a route out of Old Trafford with Los Blancos the most interested party.
United rejected a bid of £27.4million plus James Rodriguez for Pogba last week, according to The Times.
And that offer from the Spanish giants has led United to believe that Real cannot afford their talisman, report BBC Sport, especially with Bale staying put.
Bale and his representatives were in talks over a mega-money move to China which could have seen him earn over £1m-a-week but the deal broke down at the eleventh hour.
Solskjaer wants to head into the new season with players who buy into his project, and the Norwegian welcomed third summer signing Harry Maguire on Monday.
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Man Utd transfer news: Gareth Bale and Paul Pogba pitting wits in 2015 (Image: Getty)
Man Utd transfer news: James Rodriquez was offered United in a rejected bid (Image: Getty)