The Old Trafford club had seemed set to sign Tottenham Hotspur's unsettled Toby Alderweireld, but that move has run into complications, partly due to the Belgian's high demands.
United have thereby been minded to look elsewhere and, while Raphael Varane remains a long-term target as they keep an open dialogue with Real Madrid due to a number of cross-over interests - most recently the potential sale of Anthony Martial - Mourinho has increasingly been impressed by Slovakian international Skriniar.
How will Manchester United line up this season?
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One big issue is a familiar one to United, though. Inter want €80m for the 23-year-old, and have so far refused to even sit down with Barcelona.
United have encountered similar problems dealing with Serie A over the last year, as Italian clubs are highly attuned to the wealth in the Premier League.
Inter already made negotiations for Ivan Perisic last summer fraught, and Lazio now want a similar sum - around €85m - for Sergej Milinković-Savić, who remains a United target.
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