The 26-year-old, capped 75 times by Germany, will finalise his move once the formal registration processes have been completed, Arsenal's website said.
Mertesacker will join left-back Andre Santos in North London after the Brazilian completed a move from Turkish club Fenerbahce earlier on transfer deadline day.
Wenger, under pressure to strengthen Arsenal's squad after picking up just one point from three Premier League matches, also signed South Korea striker and captain Park Chu-young from AS Monaco on Tuesday.
"Santos is a quality player, with the ability to both defend well and get forward and help with attacks," Wenger told Arsenal's website.
"He has experience in the Champions League and for Brazil and has proven he can deliver at the highest level. We welcome Santos to the club and look forward to him having a big impact."
Santos, 28, made 52 league appearances for Fenerbahce and earned the first of his 22 Brazil caps at the 2009 Confederations Cup in South Africa.
Wenger does not usually take part in frantic deadline day dealing but after the 8-2 defeat by Manchester United on Sunday and the sales of Cesc Fabregas and Samir Nasri the Frenchman has moved to add proven quality to his squad.
Mertesacker's height and experience will be a boost to a defence that was torn to shreds by a rampant United and the German will also bring Champions League nous.
He has played 53 times in Europe for Bremen and took part in the 2009 Europa League final. He was also part of the Germany squad that reached the semi-finals of the 2006 and 2010 World Cups and the final of the 2008 European Championship.
