Lazio also hope to take Giroud to Serie A but the 33-year-old, who had been a fringe figure for much of the season, has been a regular for Chelsea since February and signed a one-year contract extension in May.
However, Chelsea have already bought Timo Werner, the second highest scorer in this season’s Bundesliga, from RB Leipzig for £47.5 million (Dh222m) and winger Hakim Ziyech from Ajax.
They are also interested in Bayer Leverkusen’s Kai Havertz, who could cost up to £90 million and who could complete a new first-choice attack.
While Willian is expected to join Pedro in leaving when his contract expires at the end of the season, manager Frank Lampard can also call upon top scorer Tammy Abraham, the in-form Christian Pulisic, Mason Mount and Callum Hudson-Odoi for spots in his front three.
Giroud almost left after being omitted and subsequently admitted he “did everything” to get a move in January, when Inter Milan wanted him, but Chelsea’s inability to find a replacement ensured he stayed at Stamford Bridge.
Since then, he has displaced Abraham and has scored five goals in his last six games, including strikes against Manchester United and Liverpool this week.
But the World Cup winner’s original desire to go in winter was prompted in part by his determination to retain his place in the France team for Euro 2020.
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That has now been postponed for a year but it offers him the same imperative to get first-team football to remain Didier Deschamps’ first-choice striker.
Giroud is France’s third-highest scorer, and has set his sights on overhauling second-placed Michel Platini, while he only needs three more caps to bring up a century.
Fiorentina have lacked a regular scorer this season, with Dusan Vlahovic and Federico Chiesa leading the way with just six in Serie A, though they did sign Italy international Patrick Cutrone from Wolves in January.
In contrast, Lazio have benefited from the prolific Ciro Immobile, who has been challenging Cristiano Ronaldo for the title of Capocannoniere as both have reached 30 Serie A goals for the season, but they have lacked much back up for the target man. They should be able to offer Champions League football next season whereas Fiorentina will not be in Europe.
Giroud told Canal Plus this week that he is galvanised by the impending arrival of Werner. “Of course it boosts and motivates us as strikers,” he said. “We try to make a coach’s decisions difficult – I am taking it as a positive thing.”
But he faces a choice whether to risk another season as a deputy, even if Lampard, who initially chose not to play with a target man, has been won over by the veteran.
The Chelsea manager said last week: “He's never given me a problem, he’s always trained brilliantly. When Tammy was playing well earlier in the season and regularly staying in the team, his professionalism was always good.
“I told him in January I wanted him to stay, and he was fantastic how he handled that. He’s continued to be that in the way he trains and the way he plays is important for us.”