Both North London clubs have been linked with the £20 million-rated Schneiderlin in recent weeks and, according to reports from Goal, the French international has informed his current employers of his wish to move on following such a successful top-flight campaign on the South Coast.
Such a declaration is likely to come as a huge blow for Southampton, with Schneiderlin - who made 33 Premier League appearances during Mauricio Pochettino's only full campaign in charge at St. Mary's last season - forming a crucial part of the Saints' vibrant and fluid midfield.
Indeed, this loss - if indeed it did occur before the close of the summer transfer window - would only serve to exacerbate the sense of uncertainty that is currently surrounding Southampton following the defection of Pochettino to Tottenham and the respective sales of Rickie Lambert, Luke Shaw and Adam Lallana.
Lambert joined boyhood idols Liverpool prior to the start of England's disappointing World Cup campaign, while teammate and compatriot Lallana also completed a move to Anfield earlier this week.
Left-back Shaw, meanwhile, finally ended months of speculation over his future by joining Manchester United on a four-year deal.
While this news obviously represents an enormous source of disappointment for Southampton fans and recently-appointed Dutch manager Ronald Koeman, Schneiderlin's apparent longing to seek pastures new would appear to present a real opportunity for his list of suitors.
A move to Arsenal would make a strong degree of sense for the 24-year-old, with the Gunners having developed somewhat of a proven track record for successfully nurturing the potential of several talented Frenchmen under the guidance of Arsene Wenger.
With La Liga outfit Athletic Bilbao having recently been credited with an interest in Mikel Arteta, it seems that there could well be a vacancy in Arsenal's midfield to open up shortly and Schneiderlin - who is currently out in Brazil having been called up to Didier Deschamps' senior France squad for the World Cup following injuries to Franck Ribery and Clément Grenier - would presumably be keen to link up with the likes of Aaron Ramsey and Jack Wilshere at the Emirates Stadium.
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Serie A competition
Pochettino's Tottenham also appear to be in the running for Schneiderlin's signature by sheer virtue of the Argentine's presence at White Hart Lane, however it appears that Serie A outfit Napoli - currently managed by former Liverpool and Chelsea boss Rafael Benitez - provide the major competition from the continent.
Indeed, according to Goal, the Italians already appear to have struck an initially lucrative deal with Schneiderlin's representatives, although it remains to be seen if they are yet to discuss a potential fee with Southampton.