The 19-year-old joined the Old Trafford outfit from Blackburn Rovers for £16.5 million earlier this summer, having made just 35 Premier League appearances for the Ewood Park side.
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"It's just sunk in. It took a while to sink in but I'm delighted to be here," he said in The Sun.
"It's a massive club, the best club in the world possibly, and I'm just happy to be here.
"I was away on holiday with my girlfriend when I got a phone call from the boss saying he was interested and things progressed from there.
"It was an unbelievable feeling. I picked the phone up and he says, in his Scottish voice, 'It's Sir Alex'. It's what anyone dreams of, for that phone call to happen.
"I think the lads have been brilliant and have made me feel very welcome, so I feel settled.
"I still live with my family, I'm still 19 and it's happened at quite a young age but I'm just enjoying it. They keep me grounded and make sure I stay focused."
Jones made his United debut, as did goalkeeper David de Gea, in the club's 3-1 victory over Chicago Fire at the weekend.
