Having scored 21 goals during a hugely impressive loan stint with Reims during 2022-23, Balogun has enhanced his reputation on the continent.
Despite his efforts in Ligue 1, it is accepted that the 22-year-old will leave the Emirates Stadium before the end of August, Mikel Arteta not budging on keeping Eddie Nketiah as backup to the likes of Gabriel Jesus and Kai Havertz.
That is partly due to the Gunners wishing to take advantage of cashing in on the United States international, who also has no interest in another temporary spell elsewhere.
However, although a wide array of Premier League and European clubs have been linked with Balogun, Arsenal are yet to receive close to their valuation.
Inter Milan are known to be monitoring the situation, reportedly having a change of heart over the player, while Lens are interested at a time when they need to replace Lois Openda.
Nevertheless, according to The Athletic, Monaco may look to test Arsenal's resolve, despite falling short in an effort to sign Balogun.
The report claims that Monaco have failed with verbal and written proposals presented to Arsenal's hierarchy, the former coming after the recent friendly between the sides.
While a specific figure is not mentioned, it has been alleged that Arsenal want to recoup at least £40m for a player who has made just 10 appearances in their first team.
That said, as well as rivalling the best forwards in France's top flight last season, Balogun is also a full international with the USA after switching his allegiances from England.
Monaco are still said to be holding discussions with their counterparts in North London, but it remains unclear whether they will be able to reach a compromise.
Les Rouge et Blanc are currently in the process of trying to make some major acquisitions having missed out on European qualification for next season.
Red Bull Salzburg goalkeeper Philipp Kohn and Southampton defender Mohammed Salisu have arrived at Stade Louis II, while funds have been generated from Axel Disasi's transfer to Chelsea.