Rodriguez joined Everton last summer in a £500,000 deal after leaving River Plate Montevideo and he will now be playing for Brentford for a month.
The 23-year-old will be making his debut for Brentford this weekend against Blackburn Rovers.
Despite this season not being straightforward for the Uruguayan after suffering from injuries, the Brentford boss Dean Smith has high expectations of him. Smith told the BBC, "Leandro is held in high regard at Everton and is a big prospect,"
"He likes to get in the box and thrives on crosses. He stretches the line well and those are things we need."