The 27-year-old has been allowed to leave after Copenhagen re-signed Peter Ankersen one year after selling him to Genoa.
Varela arrived in Europe in 2013 when Manchester United bought him from Penarol after an impressive trial.
After only making 11 appearances for the Red Devils he moved to Eintracht Frankfurt on loan in 2016.
Disciplinary problems led to Eintracht not renewing his loan and Varela then returned to Penarol in an attempt to kickstart his career.
Little more than a year later he was back in Europe with Copenhagen, where he has made 53 appearances.
Varela has won five caps for Uruguay, most recently appearing for his country in 2018.