The French club have even made a transfer demand that fans believe Todd Boehly will “love”.
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Barcola, who has also been linked with a move to Liverpool, can play on both wings or as a striker.
Chelsea are understood to be one of two clubs, along with PSG, to have been informed of Lyon’s demands for selling him before the window closes.
According to Fabrizio Romano, Lyon want players included in any deal for Barcola.
The Ligue 1 side are said to be keen on a player plus cash transfer and have communicate this requirement to both PSG and Chelsea.
PSG have been strongly linked with Barcola and were reportedly closing in on a £50million move earlier this month.
Fans were quick to react to Lyon’s swap demand on social media, with one tweeting: “Boehly can finally pull off his long wanted swap deal!”
Another joked that American owners such as Boehly “love” swap deals.
Barcola scored seven goals and provided 10 assists across all competitions for Lyon last season.
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The 20-year-old also grabbed two goals and two assists for France at this summer’s under-21 Euros.
Chelsea have already spent over £900million since Boehly’s takeover last year.
Their spending is set to continue as well, with a deal agreed to sign Serbian goalkeeper Djordje Petrovic.
Arsenal’s Folarin Balogun also remains a target for the Blues, who are chasing a deal for the striker.