Bristol boss Mark Sampson ( led Bristol to runners-up positions in the Women's Super League (WSL) and FA Cup this season) and England caretaker Brent Hills are among those being considered to follow former boss, Hope Powell.
Speaking at the FA Women's Football Awards, Brooking said: "We would like to be in a position early next month.
"England start their training camps in January so hopefully we will have everything in place before then."
The FA has conducted a wide search but Brooking suggested they would be looking to replace Powell, sacked in August after 15 years in the role, with someone who was also acquainted with the women's game.
He added: "I think it's important to have some knowledge of the women's game in this country but we have looked at some international people to make sure we have trawled far and wide and we have got it down to a smaller list and hope to take it on in the next few weeks."
And at the awards ceremony in London which celebrated 20 years of the FA running the women's game, Sampson's team were crowned Club of the Year for the second successive year.
Hills, who was assistant manager under Powell, has overseen England's perfect start to their 2015 World Cup qualification campaign with four wins from four games.
The 60-year-old said: "I would love to manage my country. I was very privileged to do the job for four games."