The Daily Mail says the north London rivals have both made contact with the defending French champions and are keen to get any business done before the end of the month, rather than wait until the summer.
The 25-year-old has matched the form of his team this term, by failing to match the 25 goals in 36 appearances which propelled the unfancied outfit to the Ligue 1 crown.
However, Sow's six goals this season have caught the eye of the Premier League's leading sides, as well as Newcastle and even Fenerbache.
Reports in France suggest Lille are desperate to hold on to the prize asset as they look to maintain their position at France's top table. Should either Arsenal or Spurs stump up £20million, though, their hand may be forced.