Tottenham have been linked with Barkley over recent months, with his contract at Everton due to expire at the end of next term.
There is a feeling that if Barkley does not agree to new terms at Goodison Park, Everton could be forced to cash in on the attacking midfielder.
And Redknapp believes that the White Hart Lane crowd could get very excited if Spurs did manage to snap up the England international.
“That would be some signing! That would whet the appetite of the Spurs fans,” Redknapp said.
“Everton have got to try and keep hold of him, it’d be a big problem for them to lose Lukaku and Barkley, but that would be some signing for Spurs if it comes off.”
According to the Daily Mail, Tottenham boss Mauricio Pochettino is keen to move for Barkley.
The Argentine manager reportedly is a big fan of Barkley, and would like to add him to his Spurs squad.
Tottenham look likely to be able to offer Barkley Champions League football next season, with the Lilywhites currently occupying second spot in the Premier League table.
Barkley has scored four goals and created seven assists in 27 Everton league appearances this season.