The Blues were rebuffed last week after making an initial approach understood to be worth in the region of £3m.
And Brendan Rodgers has reiterated Celtic are not looking to sell their 34-year-old shot-stopper.
The former Liverpool boss even suggested after yesterday’s 4-0 win over Hearts that Gordon will be handed a new deal if he stays.
Craig Gordon has shone for Celtic since his arrival in 2014
I really want him here, get him to a new deal
Rodgers said: ”I really want him here, get him to a new deal and then we can keep continuing and move the whole club forward.”
But according to the Daily Mail, Celtic are now weighing up Chelsea’s increased bid, with Gordon keen to assess his options.
The ex-Sunderland keeper has been lined up by Antonio Conte as a direct replacement for outgoing Asmir Begovic, who Express Sport understands is set to join Bournemouth before tomorrow’s deadline for £10m.
Brendan Rodgers doesn't want to lose Celtic's first-choice keeper as he will struggle to replace him
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Mukhtar Ali (Vitesse Arnhem)
Craig Gordon has become Chelsea's No 1 target to replace outgoing Asmir Begovic
And Gordon wants to listen to what Chelsea have to say over his future.
The Scotland international is also said to be awaiting the details of a potential new contract at Celtic, with the club aiming to up his pay.
With less than 48 hours to go until the window closes though, Conte will want a decision made quickly over a replacement for Begovic, and Brighton’s David Stockdale is also on his radar.