Yorkshire coach Gillespie is regarded as one of the favourites to take over from Peter Moores, who was sacked over the weekend after a less than successful second spell in charge.
No approach has yet been made to the county, according to director of cricket Martyn Moxon, but Brooks believes the Australian would make for an ideal appointment.
Paul Farbrace will be in charge for the two-Test series with New Zealand, which start later this month, but a new appointment is expected to be made soon after, with a bid to regain the Ashes first up.
“We don’t want to see all of our hierarchy go off??to be in charge of England and it will be sad if we lose him but he would??perfectly suited to international cricket,” Brooks said in the Independent.
“He gets the best out of very talented players, He’s been perfect for us,??changing round the fortunes of our club, and the qualities he has as a coach can??work with international players, perhaps even more so.
“Having come from an Australia team that was the best in the world for so??long, all he knows is winning and he brings a winning ethic.”
Gillespie has been in charge of Yorkshire since 2011, helping the county return to Division One in his first season and winning the County Championship title last summer.
Brooks, who joined Yorkshire from Northamptonshire in 2012, said his game has??improved immensely under Gillespie and believes the national team would??similarly benefit playing under the 40-year-old.
He added: “He is a bowler, he knows how bowlers work, he knows when you??need an arm round the shoulder or a kick up the backside. My game has gone up so??much since I joined Yorkshire.
“If you can get people in a happy place, if they are relaxed when they take??the field, they are capable of playing to their full potential. You don’t want??to be stressed or tense or worried about what you are going to do.
“It can work with international players even more. If you look at the current??England side, they are a very talented and exciting group of players. You just??need to get them playing with a bit more freedom and enjoyment.”