Both departures have left Enzo Maresca with holes to fill just days before the Premier League season starts.
Manchester City are hopeful of wrapping up a separate deal for Lille's deep-lying midfielder Ayyoub Bouaddi, valued at £85m, in the coming days, while the Daily Mail reports Koné — who featured five times for France at the World Cup — is also on the club's shortlist before the window closes.
The 25-year-old has spent the past two years in Serie A, scoring four goals in 63 appearances for AS Roma, and is more renowned for his dynamism and work off the ball than his end product. Roma have set an asking price of around £51m for Koné, broadly similar to the fee Al-Qadsiah paid City for Reijnders earlier this week.
Both Chelsea and Manchester United have previously been linked with the former Borussia Mönchengladbach midfielder.
Manchester United had spoken to his representatives last month, but their focus has since shifted towards Brighton's Carlos Baleba. Koné, who has 19 caps for France, started four of his country's matches in North America over the summer and came on at half-time in their World Cup semi-final defeat by Spain.
City remain interested in Chelsea's Enzo Fernández, with the west London club's soft deadline to complete a deal worth £120m having passed without a formal bid, though Fernández is understood to want to link back up with Maresca.
City are unwilling to reach Chelsea's demands for now. Newcastle are also exploring a move for Nico González, while City are in advanced talks to sell both Savinho and Omar Marmoush to Tottenham in a double deal likely to be worth upwards of £140m, with Maresca recently suggesting both players were considering their futures after limited minutes last season.
City's midfield rebuild follows a summer of significant turnover in that area of the squad, with Rodri's exit to Barcelona in particular representing the loss of a player who was central to the club's success under Pep Guardiola.
Maresca, in his first summer in charge, faces a tight turnaround to have any new arrivals settled before the Premier League campaign gets under way this weekend.