Monreal, 33, leaves after six years at the club to join La Liga side Sociedad on a two-year deal for an undisclosed fee.
The Spanish left-back, joined Arsenal from Malaga in January 2013 and went on to make 251 appearances for the club, scoring 10 goals.
"During his time with us, Nacho played in his natural left-back position and also in central defence. He won three FA Cups, three Community Shields and played a big part in our run to the Europa League final last season," said a statement from Arsenal.
"We would like to thank Nacho for his contribution to the club and everyone at Arsenal wishes him the best for his future."
Meanwhile, Egypt international midfielder Elneny moves to the Turkish Super Lig on a season-long loan.
The 27-year-old joined Arsenal from FC Basel in January 2016 and went on to make 89 appearances in all competitions.