"We ask that everyone kindly respect Lea's privacy during this devastating time," a rep for the actress told PEOPLE.
Following his decision to enter rehab for substance addiction in late March, Michele, 26, stood by her man.
"I love and support Cory and will stand by him through this," she told PEOPLE at the time.
After a month-long treatment in April, the costars and real-life couple reunited.
At the time, Monteith Tweeted: "Sending out big love to everyone. thank you for the continued support! It means the world to me! :)"
Michele also chimed in on Twitter, saying, "Today is a great day :) Love you all! Xo."
In May, Monteith and Michele were photographed vacationing in Puerto Vallarta, Mexico, together.
With reporting by AILI NAHAS