The Premier League champions have turned to the France international after abandoning talks to sign Edin Dzeko from AS Roma due to excessive wage demands and the length of contract requested.
Tuesday's (30 January) edition of L'Equipe reports Chelsea tabled an offer of £15m plus future bonuses for Giroud before going on to reach an agreement with the player over a move to Stamford Bridge.
That offer was dependent on a stringent deadline, however. The report states that by the time Arsenal responded to the Blues' offer, it had been taken off the table. Arsene Wenger's side had not rejected the bid at this point.
Chelsea's offer is said to have been submitted on Monday morning, during a time when Arsenal were preoccupied with talks to bring Pierre-Emerick Aubameyang to the club from Borussia Dortmund.
L'Equipe add talks between Chelsea and Arsenal over Giroud did reopen late last night, with the Blues aware of the delicate situation they find themselves in. Antonio Conte's side are now running out of time to bring in a striker to ease the strain on Alvaro Morata and having been spurned in their pursuit of Dzeko, they are now running out of viable options.
Giroud meanwhile is said to now be prioritising a move to west London.
Arsenal, Chelsea and Borussia Dortmund all find themselves in a complicated transfer stand-off with less than two days of the January window remaining.
Dortmund have agreed terms to sell Aubameyang to Arsenal, but want a replacement before signing off on a deal. Chelsea's Michy Batshuayi is now their preferred target to fill the void, with the Belgium international allowed to leave Stamford Bridge on loan this month.
But before that can happen, Chelsea want a striker to replace Batshuayi. But a failure to agree terms with Arsenal over Giroud threatens to throw a spanner in the works for all three deals, with time running out.