Giroud was desperate for a move to work with Antonio Conte once again and he had agreed personal terms on a two-and-a-half-year deal worth an initial €4 million per year, a figure that would drop to €3.5m for the following two seasons.
Both clubs stood firm on their final offers, however, with Chelsea asking for €7m (£6m/$8m) for the Frenchman, who has just five months remaining on his current deal, while the Italian giants were only willing to go to €5m (£4m/$6m) with add-ons.
The Blues could now opt for a last-minute loan to boost Giroud's chances of making France's Euro 2020 squad, with head coach Didier Deschamps having urged him to leave west London to boost his international hopes.
Tottenham are rumoured to be interested in Giroud but they are close to securing a loan with an option to buy for AC Milan striker Krzysztof Piatek and thus look unlikely to begin talks over a move for the World Cup winner.
Giroud's future could be further affected by Tammy Abraham's physical condition, with the England striker due to be assessed ahead of this weekend's Premier League fixture against Leicester City.
The 15-goal striker has kept Giroud out of the team at Stamford Bridge this season but suffered heavy bruising and swelling to his ankle in the recent 2-2 draw with Arsenal that saw him miss the FA Cup win over Hull City.
Lampard has left the option open of including Giroud in his squad for the rest of the season and has previously praised the 33-year-old's professionalism despite his desire to leave west London.
As for other outgoings, Pedro could leave the Blues in the next few days with Inter Miami, New York City FC and Vissel Kobe all interested.
Roma are another potential option for the former Barcelona winger although it would seem that the impending arrival of Carles Perez from the Liga champions will end their interest in Pedro.
Tariq Lamptey could also leave Chelsea either in January or the summer and has three pre-contract offers from Torino, Lille and an unnamed Premier League club with his Blues deal set to expire.
Academy striker Ike Ugbo, meanwhile, has had offers to hijack his Roda JC loan move from clubs in the Championship and Eredivisie but has opted to remain at his current club until the end of the season.