Manchester United have called off their search for a new left-back.
The 19-year-old was told this week that he would not leave on loan next season because he would fight to become Solskjaer's regular left-back.
An Old Trafford source told The Sun: "The manager has told Brandon to keep doing what he is doing and to keep his feet on the ground.
"At the same time, Ole has stopped looking for a new left-back. Brandon has done enough.
"There was an idea he could go out like Dean Henderson has, to another Premier League club or a team in the Championship.
"Ole loves Brandon's attitude and he will be here next season. The bosses think he will save them £50m."