The former Liverpool left-back fell out of favour under former manager Martin Jol, but he has been given a new lease of life since Meulensteen replaced his fellow Dutchman last month.
Norway international Riise is out of contract at the end of the season and looked set to leave the London club in the January transfer window when Jol was at the helm.
But the 33-year-old now hopes to be offered a new deal after returning to the starting line-up.
He told the London Evening Standard: "Rene has been brilliant for me. As soon as he came in, I said to myself that he is my type of coach.
"The way he takes training and the way he talks to the players, I knew that he was good. He has given me the confidence that I need from a manager to perform to my highest level. He put me back in the team and I have been in there since.
"I have always been committed to the club, even when I was not playing. The reason I wanted to leave is because I was not playing and I had to think of my career.
"I am at the age where I need to play. The Fulham board know I want to stay. Hopefully we can talk about a new deal because if Rene wants to keep me on, then I want to stay."