The 28-year-old Australian had recently lost his place in the Seagulls' team to Spanish goalkeeper Robert Sanchez.
“We have incredible talent and competition at this club for just one position," said Brighton boss Graham Potter.
“Understandably, as Australia's number one, Mat wants the chance of more regular opportunities."
Ryan, who joined the Seagulls for a club record fee from Valencia in 2017, has made 121 Premier League appearances and kept 28 clean sheets.
The Gunners are 10th in the Premier League, seven places ahead of Brighton, and are in the fourth round of the FA Cup and the last 32 of the Europa League.