Jenkinson has spent two out of the past three seasons on loan with West Ham but remained at the Emirates Stadium last term, making just five appearances in all competitions.
Bramall, meanwhile, needs minutes to develop his undoubted potential, which looks unlikely now in the Arsenal first-team.
Meanwhile, 21-year-old Bramall has yet to make a senior appearance for the Gunners since his transfer from non-league Hednesford Town.
Bramall is yet to make his competitive debut and will look to gain experience with the Midlands side who now sit fifteenth in the Championship table having taken just four points from their opening four games.
Arsene Wenger said over the weekend he was willing to let some players leave in an effort to free up space in his Arsenal squad and the defensive pair have been registered in time to feature against Bournemouth in Tuesday's Carabao Cup second round tie.
Deportivo La Coruña are also closing in on Lucas Perez on a loan deal with an option to buy next year.
After four games of the new campaign, Birmingham have one win, one draw and two defeats on the board.