According to Marca, Mbappe was pushed in the direction of PSG by his father despite the striker's preference to play for Real Madrid.
The move for Mbappe comes just a week after PSG shattered the world transfer record to sign Neymar from Barcelona for £198million.
Should the Mbappe deal go through it would be another huge statement of intent by PSG as their owner Nasser Al-Khelaifi bids to bring a Champions League trophy to the Parc des Princes.
Al-Khelaifi rejected suggestions that Neymar's arrival would infringe on UEFA's Financial Fair Play rejections on Saturday and told critics of the transfer to 'go have a coffee'.
However the Qatari businessman is set to face further questions about Financial Fair Play should his club complete the £163million move for Mbappe.
Mbappe courted interest from Madrid and Barcelona this summer following a stunning breakthrough season with Monaco, which saw him earn a call up to the France national team at the age of just 18.
Madrid believed they had agreed a £160million deal to sign Mbappe last month after manager Zinedine Zidane allayed the 18-year-old's fears that he would struggle for playing time at the Bernabeu.
Arsenal and both Manchester clubs had also pursued Mbappe this transfer window but PSG now look to have stolen a march on their rivals in the race to sign the teenager.
Mbappe featured for the Ligue 1 holders in their first match of the new season on Friday but was forced off after picking up a knock during the 3-2 home win over Toulouse - Monaco said they expect him to be fit for Sunday's trip to Dijon.
He has played just 41 league games for Monaco since breaking into their first team, meaning a £163million fee values the striker at a staggering £4million per game.
Mbappe would become the third key member of Monaco's title-winning squad to leave the Stade Louis II this summer should he join PSG.
Bernardo Silva and Benjamin Mendy moved to Manchester City for £43million and £52million respectively while Tiemoue Bakayoko joined Chelsea in a £40million deal.
Source: Daily Mail (UK)