According to reports in Portugal cited by The Daily Express, the North London rivals - along with fellow Premier League club Stoke - are keen to bring the defender back to his homeland but must first agree a deal with his current employers.
Cheltenham-born Dier has never played professionally in English football, having joined Sporting as a youngster and subsequently signed professional terms with the club almost four years ago.
His obvious talents have not escaped the attention of England's international coaches, however, with the player - who was on the bench for last night's win over Qatar at the prestigious Toulon tournament - having made several appearances for the Three Lions' youth teams up to and including the U21s.
Arsenal have established a relatively settled centre-back pairing in the form of Laurent Koscielny and Per Mertesacker, yet may be forced to enter the summer transfer market in search of defensive reinforcements with captain Thomas Vermaelen widely expected to leave the Emirates Stadium after falling out of favour under Arsene Wenger.
Dier's versatility - he can also occupy a deep-lying midfield role - may also appeal to Arsenal, with Wenger occasionally lacking a necessary defensive presence in that particular area.
Tottenham should also be on the lookout for at least one new defender this summer, with the club - who eventually qualified for the Europa League following an eventful 2013/14 campaign under Andre Villas-Boas and Tim Sherwood - having been dealt several humiliating hammerings by their closest rivals last season.
With doubts continuing to persist over the capabilities of the likes of Michael Dawson and indeed the happiness of the previously impressive Jan Vertonghen, it could be that Tottenham have to wait until a permanent successor to Sherwood can be sought before making improvements.
Stoke, in contrast, enjoyed an impressive first season under Mark Hughes, yet you still imagine they might find it difficult to compete sufficiently with the likes of Arsenal on the transfer front.