The 28-year-old centre-back made the temporary switch to the Serie A outfit in January with a view to a permanent move, after falling out of favour at parent club Valencia due to an argument with then head coach Miroslav Djukic.
Rami, who had been training with Milan since October, has since made six league appearances for Milan, and paid tribute to coach Clarence Seedorf and his team-mates for helping him make the transition.
"I went four months without playing and that was a tough period for me," he told the club's official website. "I've worked hard to be ready and I have a strong character.
"Credit as well to my team-mates who always gave me confidence and the coach too. It's very important for a player.
"My dream is to stay here and do all I can for the team. I think it's a dream for any player to be here."
The France international was among the scorers as Milan triumphed 2-0 at Sampdoria on Sunday.
It was the second time Rami has notched in Serie A since his switch and, though he is happy to contribute in attack, he insists his main focus is keeping the opposition at bay.
"The priority for me is to defend at the back rather than score," he added. "Of course, I'm also big and strong which helps to get the ball in the back of the net as well. I want and I have to continue to do well for this club. I want to give 100 percent."
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