The England midfielder has endured a tough campaign at Goodison Park under Ronald Koeman.
Everton's manager, who arrived in the summer, has not favoured Barkley for much of the season and openly criticised the 23-year-old.
Toffees supporters have high expectations for their academy product.
A move would be the best thing for [Barkley]
Steven Gerrard
However, Gerrard believes Chelsea or Spurs suit the playmaker more long-term.
"If he was to go to a Tottenham or a Chelsea he would fly and all the Everton fans would say why did we sell him?" Gerrard told BT Sport.
"I think from a selfish point of view, maybe for him a move would be the best thing for him.
Every Premier League club's 2016 rated by hit or miss
Fri, December 30, 2016Click through the gallery to see whether Premier League clubs have had a hit or miss 2016
Click through the gallery to see whether Premier League clubs have had a hit or miss 2016
"He's 23 and still has 10 quality years ahead of him."
Antonio Conte is in the market for new additions in January, having sold Oscar to Chinese club Shanghai SIPG for £60million.
Meanwhile, Express Sport understands Barkley will be allowed to leave for £30m with Tottenham interested, making a switch to White Hart Lane a possibility.