The 34-year-old has signed a deal until the summer of 2007 with the Glasgow giants after his contract with Manchester United was terminated by mutual consent last month.
Several Premiership clubs notably Bolton Everton and WBA were hoping to entice the midfielder to remain in England.
But Keane insists he is at the club where he 'belongs'.
Keane told Sky Sports News: 'I spoke to Madrid up as far as Tuesday and Madrid explained to me they were looking to wait another week or so.
'That was fair enough and I gave every club a chance but I thought it was about time I made a decision.
'Having spoken with the manager (Gordon Strachan) I thought this is the club for me. I have met one or two of the players and feel this is where I belong.'
He added: 'It has been a strange few weeks but this is a great move for me.
'There has been a lot of speculation in recent weeks over where I was going and I spoke to a lot of managers. It was important I took my time and made the right decision.'
But Keane has assured the Celtic fans he still possesses a fierce desire to win trophies with the Hoops.
He said: 'It is whole new challenge. It has been documented that I have been up to a few games in recent years.
'I have always enjoyed watching those games and they are seeking to improve.
'I have not come up here on an ego trip or to unwind. I am here to win matches.
'I am a team player despite a lot of reports. I feel I can help the club move on to the next level which is winning trophies.
'That is what I am in the game for. I am not necessarily in the game to be popular.
'I like pushing myself and the people around me hard but there seems to be a fault with that in the modern day.'