Several leading clubs have been credited with interest in the French midfielder, including Inter Milan, who are said to have made approaches in previous transfer windows.
Paris Saint-Germain and Atletico Madrid have also been linked with the 25-year-old, but Arsenal have now emerged as the most concrete suitors.
Last week, Corriere della Sera reported that Arsenal had entered the race for Kone, with AS Roma said to value the midfielder at around €50m.
New reporting from the same Italian outlet now suggests Arsenal have found an opening in the chase for the player.
According to the reports, Kone turned down a move to Atletico Madrid, while Paris Saint-Germain have so far failed to make their interest concrete.
Arsenal are understood to have opened talks with the player's representatives, and are reported to have already agreed personal terms.
Roma face their own pressure in the situation, with the Serie A club needing to balance their books before the end of June in order to comply with UEFA's financial regulations.
That deadline is understood to be strengthening Arsenal's position as they look to finalise the structure of a formal offer for the midfielder.
Kone has developed into one of Serie A's most complete midfielders during his time in Rome, combining physical presence, ball-carrying ability and defensive discipline.
Arsenal manager Mikel Arteta is said to view the Frenchman as an ideal option to share the defensive midfield workload currently carried largely by Declan Rice, whose responsibilities both in possession and out of it have grown significantly over recent seasons.
That defensive burden has occasionally left Arsenal short of variety when it comes to moving the ball quickly through central areas, an issue Arteta is understood to believe Kone's profile could help address.
Kone is currently away with France at the World Cup, where competition for places in midfield has been intense following the tournament's opening rounds.
Roma's situation adds further context to the pursuit, with the Serie A club needing to generate a substantial profit from player sales this summer to satisfy financial fair play requirements set out by European football's governing body.
Kone's value on Roma's books is understood to be considerably lower than his market valuation, meaning any sale at or near the figures being discussed would represent a significant financial boost for the Italian club.
The next step in any deal will be turning the personal agreement into a formal bid that satisfies Roma's valuation and their financial timetable before the end of the month.