Mbappe has just six months left to run on his contract and was the subject of a big-money bid from Los Blancos in August, but PSG rejected that interest.
The 23 year-old says he has no intention of jumping ship in mid-season, however, and is focused on winning more silverware in the French capital.
"No, I'm not joining Real Madrid in January," Mbappe said. "In January it's not happening.
"I'm in Paris Saint-Germain, I'm really happy, and I will 100 per cent finish the season as a PSG player. I'll give everything to win with Paris all titles this year.
"I will give everything I have to win the Champions League, the league and the cup. And to give all the pleasure to the fans, because they deserve it.
"I'm happy to stay. It's my city too. I'm French. I want to win everything this season."
The Ligue 1 giant is hopeful France striker Mbappe will agree a contract extension but risks losing him on a free transfer should he refuse a new deal.
PSG and Madrid meet over two legs in the Champions League round of 16 and Mbappe assured fans he remains fully committed to helping his current employers overcome prospective ones.
Despite significant spending on transfer fees and wages, PSG has failed to win Europe's premier club competition, going closest in 2020 when it was beaten 1-0 by Bayern Munich in the final.
"[The only thing on my] mind is to beat Real Madrid in February and March," Mbappe said. "We have to be ready. It's time.
"It's the most important part of the season. Of course, we want to step up now. It's been two years [that we've been in the] final [or] semi-final, but now we want to win."
Mbappe also revealed the emotional pull of playing for the capital club as a Parisian, as well as expressing his delight at this season's unexpected opportunity to link up with seven-time Ballon d'Or winner Lionel Messi.
"Paris is my city, it's where I was born, it's where I grew up," Mbappe said. "To play for PSG, it's an amazing feeling to play with your family and friends alongside of you.
"It's a big pleasure for me to say to my kids, to my friends, I play with [Messi]. We have to enjoy seeing him in Paris. It's an amazing moment in the history of the game."