Bailly is due to have a medical this week and terms are thought to have been agreed with the 22-year-old, who will cost the Red Devils about £30m.
He has played 40 times in La Liga, the first five of which were for Espanyol.
Bailly joined Villarreal for £4.4m in January 2015 and played in every game as Ivory Coast won the Africa Cup of Nations just weeks later.
It had been thought Zlatan Ibrahimovic would be Mourinho's first signing since the Portuguese succeeded Louis van Gaal as United manager.
The Sweden striker, 34, is a free agent after leaving Paris St-Germain and hinted at the weekend he would make an announcement about his future on Tuesday.
But he has refused to elaborate, saying: "There is no confirmation. Keep wondering. I am enjoying the stories."
Bailly's reputation has soared this season, which saw him play a key role as Villarreal sealed Champions League qualification, and it's no surprise to see him make a big-money move this summer.
His prime asset is his extremely impressive physical presence, with his powerful aerial ability complimenting his pace and strength in the tackle.
The brave and determined 22-year-old has less than 50 league games under his belt, though, so United fans should perhaps not expect him to immediately become an automatic starter.
He can sometimes concede possession in dangerous areas by making bad decisions, and he doesn't always show composure under pressure, but Bailly possesses all the physical tools and the rest should come with more experience.
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Exports fell 2.1% compared with October, German's Federal Statistical Office reported, while imports rose 1.5%.
Meanwhile, factory production fell by 0.1% from October as energy production fell and construction activity decreased.
The latest news follows weak industrial order figures this week.
"Things are certainly not rosy," said Bankhaus Lampe economist Alexander Krueger. "The geopolitical situation, especially the Russia conflict and the related economic uncertainty, is limiting growth,"
On a seasonally adjusted basis, the country's trade surplus shrank to €17.7bn (£13.9bn) in November.
Germany exported goods worth around €95.8bn and imported goods valued at €78.0bn in the month.
In December, Germany's economic affairs ministry said the country's recovery was "progressing at a slow pace."
This is a disappointing batch of German data. It pours cold water on hopes that the Eurozone's traditional economic powerhouse will drag its neighbours out of the gloom in the near future.
The figures don't necessarily point to a renewed contraction in the German economy.  But they are consistent with no significant acceleration from the feeble 0.1% growth that Germany recorded in the third quarter of last year, following a contraction of the same magnitude in the previous three months.
There were also declines in French and Spanish industrial production, adding another layer of gloom to the picture.
These figures provide a little extra support to the view that the European Central Bank will embark soon on a major programme of quantitative easing - buying financial assets notably government debt with newly created money.
Villarreal central defender Eric Bailly is set to be Jose Mourinho's first signing as Manchester United manager.
[NEXT_CONCEPT]
German exports and production fell in November compared with the previous month.