Officials from the Serie A side held talks with the French striker’s agents last week.
There is less than £1m (€1m) between Chelsea’s request and Inter’s offer. The Blues have asked for £4.3m plus £850,000 (€5m + €1m), while Inter have offered £3.4m plus £850,000 (€4m + €1m).
There is an ongoing exchange of emails and there is a desire to close the deal as soon as possible.
Lyon have also been keen to sign the 33-year-old forward, while Aston Villa made a loan approach earlier in the window.
Giroud was at Stamford Bridge on Saturday watching the 3-0 win over Burnley from behind the team’s dugout.