Giroud, 32, is the Europa League's top scorer with 10 goals, but has started only seven matches in the Premier League this season. "Next year I will need to have a more important role to play in the team," the striker demanded on Wednesday.
However, Giroud's ultimatum may ring hollow due to the small print in his contract.
"He (Giroud) is sure (to stay) because the club has an option for another year, another season. I spoke to the club two days ago and we agreed to exercise the option. So, he will remain," Sarri said, as reported by The Telegraph's Sam Wallace.
Alvaro Morata was previously the biggest culprit of taking minutes from Giroud in the middle of Chelsea's three-man strikeforce. By the time Morata was loaned to Atletico Madrid near the end of January, Gonzalo Higuain had already been drafted in on a temporary deal from Juventus.
Giroud has started only one league match - when he scored in a 3-0 ease past Brighton & Hove Albion - since Higuain's arrival.
Chelsea round off the Premier League campaign with matches against Burnley, Manchester United, Watford, and Leicester City. The first leg of their Europa League semfinal meeting with Eintracht Frankfurt will be contested in Germany on May 5.