Brazil goalkeeper Julio Cesar has joined Benfica on a two-year deal after being released by QPR.
Riether has impressed since moving to Craven Cottage on a year-long loan from German side Cologne at the start of the season, with boss Martin Jol previously indicating his desire to keep him.
Jol is keen to trigger a clause in his contract to secure a permanent move and Riether, who scored in the weekend's 2-2 draw at Sunderland, admits he is enjoying life in west London.
"I'd like to remain here," he told the club's official website.
"Fulham have an option to buy me and so I think in the next few weeks there will be talks about my future.
"At the moment I can't really say what will happen.
"But I like it here and I'm thankful that Fulham gave me the chance to play in the Premier League."
Riether had never played outside of his home country until making the move to Fulham.
But the 29-year-old, who made his two Germany appearances in 2010, admitted he had felt at home straight away.
"In my mind, I've never thought of myself as being here 'on loan'," he said.
"I give my best as if I'm here for a long time. I've never felt I was coming here for a short time, for a holiday.
"I always give everything in each game I play and see myself as a Fulham player, not a Cologne player.
"For me, it's been a great time here so far. It was my first experience of another country and it was a big step for me to come to England.
"I have always played in Germany but this was an exciting challenge for me.
And I have to say Fulham is like a family and I enjoy playing here. The fans are great, I like the stadium and it's been all good so far. I feel I have adapted well but that is down to everyone at the club too."
Riether is one of a number of Fulham players who could leave at the end of the season should a deal not be sorted out.
Jol admitted last month that 11 players' contract were close to expiring, although since then Damien Duff and Philippe Senderos have signed one-year extensions.
"We have 11 people who are at the end of their contract (at the end of the season)," Jol said. "So we have a few problems."
Veteran goalkeeper Mark Schwarzer said last month that he had begun discussions over a new deal while Brede Hangeland is yet to commit himself beyond the end of the season.