Giroud signed an 18-month deal when he joined from Arsenal in January 2018, but an option was included to extend it by another 12 months.
The France international is the joint-top scorer in the Europa League this term with 10 goals, but has expressed his unhappiness about a lack of starts for Chelsea in the Premier League under Sarri.
Nice, Marseille, Bordeaux and Lyon have all expressed an interest in the 32-year-old, but Chelsea appear to have convinced him to stay.
Chelsea are under pressure to keep as much of the squad together as possible because they have been given a two window transfer ban by FIFA due to breaching rules over acquiring foreign minors.
They have appealed to the Court of Arbitration for Sport but their case is not expected to be heard for some time.
It is understood Chelsea have not asked for the punishment to be delayed while the process is carried out yet, however CAS can decide to do that of their own accord.