HITC Sport understands Arsenal are one of a number of clubs considering an offer.
The 17-year-old sensation is already a member of the first-team with the Principality outfit, having started five games this season and bagging three goals.
He is being likened to former Monaco stars Thierry Henry and Anthony Martial, although HITC Sport understands that Arsenal boss Arsene Wenger has been aware of the player for a number of years.
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However, his performances have now seen him emerge as a target for a number of Arsenal's Premier League rivals.
Manchester United's Anthony Martial celebrates scoring
Tottenham and West Ham have also been checking on him, whilst the likes of Juventus and Atletico Madrid have already compiled dossiers on the player - who many in French football believe could go on to become a senior international.
A well placed French source told HITC Sport: "The amount of English clubs coming to watch him is growing every week and despite only being 17, this could be his only full season at Monaco if his progress continues as it is."
The English clubs are now looking closely at Mbappe, as they realise it would already take over £20million to get him and those costs could easily double in the next 12 months - like it did with Martial.
Thierry Henry celebrates goal v Manchester United 2000
Mbappe, like Henry and Martial, is comfortable anywhere across the forward line.
He was raised on the outskirts of Paris and was a star student at France's famed Clairefontaine academy and was wanted by every top club in the country, but he opted for a move to Monaco, where he has risen rapidly through the ranks.
In February this year he beat Thierry Henry's record to become the youngest goalscorer in the club's history. Monaco have already moved to tie him down to a deal until 2019 - but they could not resist a sizeable offer.
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