However, a fee between the two clubs remains unresolved.
Manchester United have already added Youri Tielemans from Aston Villa and Andrey Santos from Chelsea to overhaul Michael Carrick's midfield, but see Koné as the more defensively-minded profile they still lack. The France international impressed at this summer's World Cup and has emerged as United's leading target for that role, with the Red Devils having tabled an offer of around €50 million. Roma, however, are said to be holding out for closer to €60 million as they look to balance their books before the transfer window closes.
United also face competition for Koné's signature. Chelsea, Atlético Madrid and Saudi side Al-Ahli have all been credited with interest in the 25-year-old, adding pressure on United to move before Roma's asking price climbs further.
Complicating matters further is Koné's own preference. According to Il Tempo, the midfielder's genuine dream is a move to Paris Saint-Germain, and he would have no hesitation choosing Luis Enrique's side over Old Trafford if PSG were to match Roma's valuation.
The Ligue 1 champions have shown interest in strengthening their midfield rotation but have not yet made an approach, leaving United as the frontrunners for now.
United's pursuit of Aurélien Tchouaméni remains on hold in the meantime, contingent on Real Madrid completing a deal to sign Manchester City's Rodri, which would free up a midfield slot at the Bernabéu and potentially open the door for United to test Real Madrid's resolve over their France international.
Koné's rise this year has been striking. Having joined Roma from Borussia Mönchengladbach, he became a regular under Gian Piero Gasperini and forced his way into a deeper role for France during the World Cup, drawing praise from pundits for his ball-carrying and defensive coverage.
That form has driven up his valuation considerably, leaving AS Roma in a strong negotiating position even as United push to have a deal wrapped up before their pre-season preparations are complete.