Both of those Premier League giants have found themselves heavily linked with the Brazilian playmaker in recent times but, according to the latest speculation from France cited by The Daily Star, it is the Ligue 1 runners-up who appear the frontrunners for his signature after beginning negotiations over a £50 million deal for Costa and fellow Shakhtar Donetsk man Alex Teixeira.
Costa's future at the Donbass Arena has been subject to considerable doubt in recent weeks, with the 23-year-old - who arrived at the club from Campeonato Brasileiro Serie A runners-up Gremio back in 2010 - having apparently become unsettled due to the ongoing, vicious conflict that shows no signing of abating in Ukraine.
Despite the club's failure to triumph over rivals Paris Saint-Germain last season and the subsequent departure of manager Claudio Ranieri, a move to Monaco would appear to make sense for Costa.
Although the French League has typically been regarded below the likes of Serie A, La Liga and the Premier League in terms of quality and reputation recent years, the significant sums that continue to be invested at both Stade Louis II and the Parc des Princes have resulted in the attraction of a number of world football's star names.
While the recent departure of Colombian sensation James Rodriguez to Spanish giants Real Madrid will have come as a real blow to Monaco and Ligue 1's global standing in general, it does notably open up a place in new boss Leonardo Jardim's midfield.
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Confusing Premier League interest
It is currently unclear as to whether either Manchester United or Arsenal plan to disrupt discussions between Monaco and Shakhtar with a concrete offer of their own for Costa.
In truth, a sustained and expensive pursuit of the player appears to make preciously little sense for either club.
New Manchester United manager Louis van Gaal - who has so far guided his new charges to a three-match unbeaten streak on the club's annual pre-season tour of the United States - is certainly not short of options in attacking midfield, with the likes of Shinji Kagawa expected to continue to suffer from a lack of regular playing time as it is.
Arsene Wenger has a well-documented enthusiasm for accomplished attacking players who are comfortable with his particular brand of football, yet Arsenal also have a number of names currently at the club who are able to fill Costa's position with relative ease.
A move to Monaco, therefore, could be best for all parties concerned.