Catt, now player-coach at London Irish, told BBC Radio Five Live: "I would love to work with 'Johnno', he's got a wealth of experience in the game.
"He was a fantastic rugby player and he's got the utmost respect from a lot of players and leaders in the game.
"When he says something, you do it, or you suffer the consequences. He knows exactly what he wants to achieve - and he makes sure it happens."
South Africa-born Catt, 36, was a key member of England's 2003 World Cup-winning squad in Australia under Johnson's captaincy and a member of the England squad which reached last year's final in France.
Johnson's former Leicester and England team-mate Austin Healey and Northampton coach Jim Mallinder are other contenders for the role.
Meanwhile, Brian Ashton, until Wednesday England's head coach, is considering his response to the Rugby Football Union's offer of a post as national academy boss.