The 25-year-old officially became a free agent in the summer after running down his Blaugrana contract, but he soon agreed to a two-year extension.
Dembele has now entered the final 18 months of those terms, potentially leaving Barca in a position where they could be pressured to sell in the current or future transfer windows.
Barcelona unsuccessfully tried to force Dembele out of the club last January, and he was widely expected to complete a mid-season switch to PSG, but a move failed to materialize.
Chelsea and Manchester United were also understood to hold an interest in the former Borussia Dortmund man, but the English giants are now said to have pulled out of the running.
According to Spanish outlet Fichajes, Mbappe and Dembele have forged a strong relationship in the France camp, and the former believes that the Barcelona man would be a good signing for PSG.
Dembele was a constant fixture of the France team during their run to the World Cup 2022 final, setting up two goals in seven games, but he was taken off in the first half of the showpiece match with Argentina.
The report adds that Barcelona boss Xavi has been left delighted with Dembele's contribution to Barcelona this season, and he is keen to see the ex-Rennes starlet stay at Camp Nou for longer.
However, Dembele could supposedly be allowed to leave the club in the current window given his contract situation, although Barca are expected to prioritize renewals over signings in the New Year.
The Frenchman has shone for Barcelona this season with five goals and seven assists in 21 matches across all competitions, having ended the 2021-22 season with a league-high 13 assists in La Liga.
Dembele now has 37 goals and 41 assists under his belt from 171 games in a Barcelona jersey since arriving in a €140m (£124.1m) deal five years ago, winning two La Liga titles in that period.
Neither Dembele nor Mbappe enjoyed the desired results with their clubs at the weekend, as Barcelona were held to a 1-1 draw by city rivals Espanyol in a chaotic derby.
Meanwhile, PSG suffered their first defeat of the season to second-placed Lens, who bested the champions 3-1 on Sunday night to close the gap at the top of the Ligue 1 rankings.
The French champions had to make do without the suspended Neymar and absent Lionel Messi - who is yet to return from World Cup duty with Argentina - as Mbappe led the line alongside Hugo Ekitike.
Ekitike equalized only three minutes after Przemyslaw Frankowski's opener, but Lois Openda and Alexis Claude-Maurice got in on the act for Lens to cut PSG's lead at the top down to four points.