He has scored seven goals and provided three assists in 16 top-flight games for Brighton this season.
"He is a versatile player with high technical ability, intelligence and a great deal of experience," Arsenal boss Mikel Arteta said.
The Gunners say they have submitted the required documents and hope Trossard will be available to face Manchester United at Emirates Stadium on Sunday (16:30 GMT).
Trossard, who made three appearances for Belgium at the 2022 World Cup, will wear number 19 for Arsenal.
"It is good work from everyone at the club to complete the signing of Leandro," Arteta told the club's website.
"Leandro strengthens our squad as we head into the second part of the season and we're all looking forward to working with him.
"We welcome Leandro and his family to Arsenal Football Club."
Trossard's agent, Josy Comhair, said last week the forward wanted to leave Brighton and would not sign a new contract after falling out with boss Roberto De Zerbi.
He was left out of the squad for the 3-0 home victory over Liverpool after leaving training early without permission.