Scotland Under-19 international McLennan joined the Royals in 2012 after leaving Gillingham.
He memorably scored a 119th minute leveller against Liverpool after going on as a substitute in the FA Youth Cup quarter-final, which Reading went on to win. He finished that season with 27 appearances and two goals and signed his first professional contract at the Madejski Stadium in July 2014.
The 19-year-old, who is from Ashford in Kent, spent time at Vanarama National League South club Hayes and Yeading last season, tasting senior first team football for the first time.
He impressed Robins boss Gary Johnson in a trial match at Winchcombe School last week and has now agreed to join Cheltenham, becoming the club's 10th new signing of the summer.