Enrique posted a message to his followers on Instagram detailing the "toughest few weeks of my life."
The 32-year-old said he has had surgery to remove the tumour and is now recovering.
Enrique retired from football in 2017 at the age of 31 after being released by Real Zaragoza. He spent five years at Liverpool between 2011 and 2016.
Enrique was part of the Liverpool side that won the 2012 Carling Cup, the team's last major piece of silverware.
The club posted their best wishes to their former player on Twitter:
Perhaps the most prosperous spell of Enrique's career came during his time at Newcastle, though, where he was a big fan favourite.
After joining the Magpies from Villarreal in 2007, he enjoyed four years at St James' Park, cementing himself as a key player. In the 2009-2010 season, he helped Newcastle earn promotion back to the Championship from the Premier League.