The Ligue 1 champions want the two men to complete an all-star attacking triumvirate alongside Neymar this season.
Mbappe’s future has been the subject of much speculation in recent months, with the World Cup winner yet to reach an agreement over a contract extension with PSG.
The 22-year-old is being heavily linked with Madrid after entering the final 12 months of his current deal, but Messi’s impending arrival at Parc des Princes could lead to a swift resolution that sees him remain in Paris for the long term.
Messi admitted that joining PSG is "a possibility" in his emotional final press conference at Barcelona, who were unable to re-sign the six-time Ballon d’Or winner for the 2021-22 campaign due to "financial and structural obstacles".
It has been suggested that Financial Fair Play rules won’t allow PSG to register three global stars, but they are expecting UEFA to be more flexible due to the financial constraints caused by the coronavirus crisis.
They have also received government backing, with the impending acquisition of Messi and renewal of Mbappe’s contract both seen as beneficial moves for French football and France as a whole.
PSG are planning to sit down with Mbappe’s representatives to thrash out an agreement after Messi is unveiled, but Real Madrid are keeping a close eye on the situation.
Real’s stance on Mbappe
Goal has learned that Real are waiting to hear the result of PSG’s negotiations with Mbappe before deciding how best to continue their pursuit of the forward.
The Blancos would like to sign Mbappe this summer, but are prepared to hold off until next year before launching a formal bid.
Real are currently in the process of trying to reduce their squad, with permanent transfers and loan deals being explored ahead of Carlo Ancelotti’s first season back in charge at Santiago Bernabeu.
Mbappe has featured in 172 games for PSG since joining the club from Monaco in the summer of 2017, scoring 132 goals.
The attacker has also recorded 62 assists while helping PSG win three Ligue 1 titles and reach their first-ever Champions League final.