Sanchez's contract with Arsenal is due to expire in June 2018 and talks over a new deal were put on hold until the end of the season after both parties failed to reach an agreement over a renewal.
Chelsea are believed to be keeping a close eye on the attacker's situation, but Campbell remains hopeful he will stay at Emirates Stadium.
"The club should do their upmost to keep Sanchez," the former Arsenal defender told Omnisport.
"Signing players who are at his level will help with that and will also inspire players to believe they can go on and achieve great things next season.
"If you do it [move to Chelsea], why are you doing it? Are you moving for football reasons or for the money? You have to ask yourself whether you fit in at that club.
"Sanchez is alone in his energy at Arsenal, but seeing other players at his level will help to push him and the club on."
The 28-year-old has been a key figure for Arsenal this term, scoring 22 goals in 39 appearances in all competitions