Arsenal are said to want at least £7m for the 23-year-old’s signature.
Palermo may consider Campbell a replacement to Paulo Dybala – who joined Juventus in a €32m (£22.5m) deal at the beginning of June.
The Serie A club are expected to return with a higher bid next week, and could be prepared to meet the Gunners’ valuation.
Campbell spend last season on loan at Villarreal, where his form hugely disappointing.
He managed just one goal in 20 appearances after a promising previous previous campaign with Olympiacos.