Shea, 21, who plays for FC Dallas in MLS, will team-up with the Gunners after playing for the U.S in friendly matches in France and Slovenia on November 11 and 15.
Dallas said in a statement on their website that Shea would then be returning to Texas in December.
"The decision to send Shea to train with the Gunners was made after U.S. head coach Jurgen Klinsmann encouraged his MLS-based players to spend the off-season training overseas to enter January camp in top shape," said the statement.
Shea told Reuters that his aim was to get some valuable experience.
"[Jurgen's] been pushing and wanting us to train in Europe in the off-season and it is good to get that experience.
"It's just strictly training and get that experience and hopefully raise my game to another level," he said.
Shea has two years remaining on his contract with Dallas but says he would relish a chance to play in England or Europe at some stage.
"I think, obviously I am happy where I am, but I want to, eventually when the time is right, try it out and give it a shot," he said.
For the moment though, the winger said he was relishing the chance to spend a month under the charge of Arsenal manager Arsene Wenger.
"To see how he does things... he has had so many young players over the years and turned them into who they became, that experience will be pretty cool," said Shea.
FC Dallas head coach Schellas Hyndman has given his blessing to the arrangement which doesn't disrupt his club's plans.
"Affording Brek the opportunity to continue to work on improving his game and measure himself among the best players in the world is something we are all excited about," he said.
"It's a winning situation for Brek, FC Dallas and the U.S. team, as the extra training time will help push his game to the next level."
There has been no suggestion that Arsenal intend to sign Shea, who usually plays wide on the left.
Arsenal have strong roots in North America, with a club scout, Danny Karbassiyoon, and the club's chief executive Ivan Gazidis being a former deputy commissioner of MLS.
MLS's off-season runs from late November to mid-March and Klinsmann has been pushing players in and around his national team squad, to take advantage of the break to train with foreign clubs.
U.S. number two keeper Bill Hamid is currently training with another Premier League club, West Bromwich Albion.
