Anthony Allen and Louis Deacon have been confirmed as club vice-captains.
Cockerill said: "We have a very good core of leaders within the playing squad which is very important during a long season when there are changes due to international calls, unavailability and injuries.
"Toby is a very experienced player at the top level with 57 caps for England and he is now starting his sixth season with Leicester. He is in a key position in the team, a key decision-maker and communicator, and he led the team very well at the end of last season.
"He is a mature player and a good influence in the squad on and off the field.
"Louis Deacon and Anthony Allen will be our vice-captains. They are both experienced players with a level of responsibility in the squad and who are well respected by their colleagues."
Flood said: "It is a great honour and a privilege to captain Leicester Tigers.
"I enjoyed the role of captain during last season and I'm looking forward to continuing in the new season.
"There are so many good players and leaders in this squad that everyone shares the responsibility on and off the field."