Valverde revealed he trained with the Gunners before being snapped up by Real.
He told his club's TV channel: "Arsenal asked about me when I was with Penarol, and I had travelled to practise with them.
“It was very small. Sharing training with Alexis Sanchez and the fact that he came to help me with the language was something very nice."
The Uruguayan midfielder is seen as the natural successor to Ballon d'Or winner Luka Modric at the Bernabeu.
Madrid swooped to land him from his local club for a paltry £4.5million two years ago.