The Daily Mirror and Daily Star both said Fenerbahce had made an eight million pound offer to lure Campbell with a three-year contract worth 70,000 pounds a week.
Campbell scored Arsenal's goal in the 2-1 Champions League final defeat by Barcelona last week but the 31-year-old has had a troubled season.
He walked out after being substituted at halftime in a home game against West Ham in February and following his return suffered ankle and toe injuries and broke his nose.
The media reports said Campbell's agent Sky Andrew was understood to have met Fenerbahce officials last week.
Campbell, who joined Arsenal in 2001 from north London rivals Tottenham, was named earlier this month in England's 23-man World Cup squad.
Fenerbahce had the Turkish title snatched from their grasp on the final day of the season last weekend when they could only draw away to lowly Denizlispor when they had to win. Big rivals Galatasaray pipped them to the championship.