It has been widely reported that the 18-year-old is a top target for Gunners boss Arsene Wenger and Real Madrid, but his future still remains up in the air as Monaco try to retain his services for at least another season.
However, according to Marca, the Premier League outfit have increased their bid to £123m, a figure which would dwarf the current world record which Manchester United paid Juventus for Paul Pogba in August.
It is unclear whether Mbappe would be more receptive of a move to the Emirates Stadium or the Bernabeu, with Arsenal able to offer him more senior action than Real with Gareth Bale, Cristiano Ronaldo and Karim Benzema all likely to remain in Spain.
Last season, Mbappe scored 15 goals in 29 outings as Monaco won the Ligue 1 title, while he netted a further six times in nine appearances during his club's journey to the Champions League semi-finals.
Other clubs hold an interest in the France international, but Arsenal and Real Madrid remain the favourites to pull off any deal.