Rodgers was one of the early frontrunners to take over from Wenger at the Emirates Stadium, having won back-to-back Scottish Premiership titles with Celtic since moving to Glasgow in 2016.
The former Liverpool manager could yet win the domestic treble for the second year in a row too, and Campbell believes that the Northern Irishman would be a good appointment for the Gunners.
"I wouldn't turn my nose up at Brendan Rodgers. I know a lot of Arsenal fans wouldn't want Rodgers in, but he has learned from his mistakes at Liverpool," Campbell told Sky Sports News.
"He has gone on to Celtic and done fantastic with them. Now he knows how to win. I don't think he would be a bad appointment if he was brought in."