Reports suggest that Brendan Rodgers has enquired about the availability of taking the Costa Rica international on loan for a season as he looks to bolster his attacking options in the wake of Luis Suarez's departure to Barcelona.
Thus far, Rickie Lambert has arrived as Daniel Sturridge's understudy, but with Iago Aspas back in Spain with Sevilla and Fabio Borini on the verge of a move to Sunderland, Liverpool are light in attack.
However, according to Goal.com, Arsene Wenger wants to keep hold of Campbell until at least January.
In a press conference before his side's 2-1 victory over Crystal Palace, Wenger told reporters: "He has come from a non-competitive world to a world with hyper-competition.
"He has adapted slowly, but has adapted very well, he has a very good spirit and attitude. He is a member of my squad at the moment, He had a good preparation and played well against Benfica.
"In training he is very focused and eager to do well. I would like to keep him."
Campbell spent last season on loan at Olympiakos, where he scored eight goals in 32 appearances.