The 50-year-old rocker hit it off with the 'Dallas Buyers Club' star when they both attended a wedding in the US recently, and the pair, along with Noel's wife Sara MacDonald, had a ''proper laugh'' together as they stayed out drinking until almost 5am.
He said: ''We were in America on tour with U2 and we were at a wedding of a really good friend of ours who lives in Hollywood.
''We were in the hotel foyer before we had to go to the wedding, me and Sara had had a couple of drinks - it was 11 in the morning, we were starting early, it was a day off - and we were like 'oh there's Matthew McConaughey, get on that'.
''I don't know a great deal about his films but I love 'Wolf of Wall Street' and I was a bit tipsy so I went over and I said, 'You don't know me, but I love you.'
''And he went, 'First of all, I want to thank you for coming over here, spreading the good word. Secondly, let's get this couple married and we're gonna stretch the... something out of this party.' and I said, 'You know what mate, we are gonna be friends forever.'
''And I wish I could tell you all the tales, but they all involve swearing and it doesn't work if you're saying 'poo'...
''He tried to get into mine and Sara's room at five in the morning, wanted to [carry on with] the minibar.
''Needless to say, he's a proper dude, we had such a laugh.''
And the 'Holy Mountain' singer - who has 17-year-old daughter Anais with ex-wife Meg Matthews, and sons Sonny, seven, and 10-year-old Donovan with Sara - found it particularly funny when the 48-year-old actor advised him never to wear a hat.
He recalled on Radio X: ''He said to me at the bar, 'My brother, do you wear a hat? You can't wear no hat, your thoughts are too big for hats.' And I went, 'That is so true'.''
Noel's revelation about his encounter with the Oscar-winning actor came after he chose Matthew to play him in a movie of his life.
Asked who'd get the starring role in his biopic, he said: ''Initially I'm thinking Morgan Freeman. Maybe Matthew McConaughey. I could see him, 'Alright alright alright'. He's got the body, he's got the looks, can't fail.''
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