According to MailOnline's Oliver Todd, the striker is eager to show he is a "loyal warrior" and fight for his place with Lacazette, who appears set to join in a £44 million move.
Per Todd, he told the Arsenal Weekly podcast:
"I won a few titles in my career, that is an important thing, but there is another thing which is important for me.
"And that is that people [see you as a] loyal warrior on the pitch—that you always give your best for the team.
"And that's why I want to carry on my good journey at Arsenal and I always want to give my best.
"I would love to win another title, or maybe more until the end of my contract. The motivation is here, always doing your best to make your family proud."
The 30-year-old has three years remaining on his current contract at Arsenal.
Despite starting just 17 of his 40 appearances last season because of Alexis Sanchez's shift to a centre-forward role for much of the season, he scored 16 goals and laid on seven assists.
The Mirror's John Cross believes the Frenchman would make a fine addition to the Toffees:
He would also be an excellent buy for West Ham. While neither side would be able to support him with as much creativity as he enjoys from his Arsenal team-mates, both teams would arguably be better able to play to Giroud's strengths than the Gunners—the Hammers in particular.
Arsenal are wise to seek an upgrade on Giroud, and Lacazette looks to be just that, even if he is behind him in the pecking order for France.
Football.London's Charles Watts believes the club should keep hold of the former despite the latter's likely arrival:
Indeed, as he did last year, Giroud has shown himself to be a superb squad player, and his height and physicality allow Arsenal a different option off the bench.
He's not likely to relish the prospect of playing second fiddle once again, but it's evident he holds the club in great affection.
If he were to leave, he would likely have his sights set on another club that typically operates in the UEFA Champions League, so it could be a struggle for either Everton or West Ham to land him in any case.