Jose Mourinho has already made two signings this summer but is keen to acquire more players in the transfer window.
Finding a new defensive midfielder is the Portuguese's priority despite Michael Carrick signing a new one-year deal.
And Mourinho is keen to bring Tottenham star Dier to Old Trafford.
United have already seen a £50m bid rejected for the England international with Spurs desperate to keep hold of him.
Manchester United are considering making a £60m bid for Eric Dier How Manchester United could look with Eric Dier Wed, July 12, 2017
How could Manchester United look if they secure a deal for Eric Dier?
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How Manchester United could look with Eric Dier
Having already sold Kyle Walker for £50m to Manchester City, Mauricio Pochettino, and chairman Daniel Levy, aren't prepared to lose another defensive player without finding a replacement.
But The Mirror claim United are preparing to test the waters again with an improved £60m offer to tempt the London club into selling.
Mourinho had hoped he could sign Dier for around £40m and an offer double his wages with the 23-year-old keen on the switch.
Jose Mourinho is on the lookout for a new defensive midfielder
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But with Spurs reluctant to sell to their Premier League rivals they are unwilling to part with him on the cheap.
The report claims United are also pursuing Chelsea's Nemanja Matic but face a similar problem with Chelsea unwilling to do business.