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Published 22:37 26 Mar 2018 BST
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Regardless of his long-term contract at Celtic, the report claims Tierney is only likely to cost United around £25m - around half the fee they would have to stump up for Rose.
The left-back position is said to be a priority for Mourinho. As well as Shaw, he is also expected to sanction the sales of Daley Blind and Matteo Darmian - two players who have also played at left-sided full-back for the club in recent years.
Blind came close to a January switch to AS Roma, a move which could be resurrected in the summer. Darmian, an Italian international, has also been linked with a move to his homeland with Juventus thought to be interested. He has also reportedly caught the attention of Valencia.
Tierney, a Scottish international, is said to have drawn favourable reports from Mourinho's scouts this season with United expected to follow this up with an offer in the weeks ahead.