According to the latest speculation from Goal, the Gunners have tabled an offer of £21.3 million for Carvalho, with Sporting said to be willing to reduce the player's rather extravagant £28 million price tag.
Indeed, the same source further suggests that Arsenal have been toiling to complete a deal for many weeks now, with the club evidently hopeful of sealing things properly prior to Monday's transfer deadline.
Carvalho's situation has developed into one of this summer's longest-running and most uncertain transfer sagas.
Linked frequently with a move to Manchester United, it appeared highly likely that the 22-year-old would be moving to Old Trafford with Louis van Gaal evidently requiring a more robust presence in midfield and his protracted pursuit of Juventus' Arturo Vidal looking increasingly futile.
A move to the 20-time top-flight champions has yet to materialise, however, with Arsene Wenger presumably keen to capitalise on his rival's hesitance before the window closes until January.
A defensive-midfielder by trade, Carvalho - who made two appearances for his native Portugal at the 2014 World Cup - enjoyed a hugely impressive 2013/14 campaign, instantly becoming one of the most important members of the first-team squad at the Estádio José Alvalade.
Arsenal currently have plenty of options to choose from in the centre of midfield, although with captain Mikel Arteta set to be sidelined for the next few weeks with an ankle injury sustained during the club's Champions League qualifying first-leg tie against Besiktas in Turkey, it stands to reason that Wenger could be looking to strengthen even further.
Manchester United to weigh in?
Whether this new offer from Arsenal will entice Manchester United into entering some sort of bidding war remains to be seen.
The capture of Angel di Maria and the imminent arrival of Daley Blind from reigning Eredivisie champions Ajax will provide an obvious boost to Van Gaal's squad, although he remains some way short of securing the same sort of strength in depth in key positions that is available to the likes of Manchester City and Chelsea.