The Premier League winners parted ways with Diego Costa and replaced him with Alvaro Morata. The Spain international arrived at Stamford Bridge from Real Madrid last summer and has struggled to impress in recent weeks.
Morata has scored 10 goals from 20 appearances in the Premier League this season. Since his goal against West Bromwich Albion on 18 November, the former Juventus striker has scored three goals in all competitions.
Antonio Conte is in the market for a new striker and Parlour has advised the ex-Italy coach to make an approach for Giroud in January. The former Gunners man believes the France international "would be tempted" to make a switch to Chelsea as that will give him an opportunity to continue living in London.
"He would be perfect. Whether Arsenal would want to lose another of their players to a rival is another matter," Parlour told Sky Sports.
"But I am sure if Chelsea put a decent bid in he would be tempted because he hasn't got to move, he can live in London, a bit like Petr Cech. He was settled in London and wanted to stay there."
Giroud has seen his first team appearances being limited since the start of the last season. Arsene Wenger fielded Alexis Sanchez to lead his side's attack for the 2016/17 season. The arrival of his compatriot Alexandre Lacazette from Lyon last summer has seen him make only one start in the Premier League this term.
Lack of playing time at the Emirates has seen him being linked with a move away from the club. The reports claim that Giroud could be involved in a player-plus-cash deal that will see Arsenal sign Pierre-Emerick Aubameyang from Borussia Dortmund.
Meanwhile, Chelsea have been linked with a move for West Ham United striker Andy Carroll, Stoke City's Peter Crouch and Tottenham Hotspur frontman Fernando Llorente.
Parlour's comments have been backed by his former Arsenal teammate Paul Merson. The latter believes that Giroud could be the "answer" to solve Conte's striker problems at Chelsea.
"It's staring Chelsea straight in the eye. They were looking at Carroll, now they are talking about Crouch - Giroud is the answer," Merson explained.
"If they are looking for someone who is big and strong and can come on and change a game I don't see anybody better. Giroud is very underrated."
The Express reports that Giroud is earning £100,000-a-week at Arsenal. Should Chelsea sign him in the mid-season transfer window, the English champions are unlikely to have any trouble to meet the striker's current wages.