The 21-year-old is a graduate of the same Tottenham Hotspur youth team as fellow new Robins recruit Jack Munns and Liverpool midfielder Kevin Stewart, who spent time on loan at Whaddon Road last season.
Having made a total of 23 appearances for Swindon Town over the past two seasons, Barthram was the subject of interest from other League clubs, but Johnson persuaded him to play in a team gunning for promotion from the Vanarama National League at the first attempt.
"I had a call from Gary Johnson saying he wanted me to join and I already knew a bit about Cheltenham, but he really sold the club to me," Barthram said.
"One of the main things that interested me was that he is clearly not intending to be in the National League for more than one year and he wants to get straight back up into League Two.
"I know what he did at Yeovil, taking them into the Championship and I think he can do something similar at Cheltenham, or I wouldn't be here. We are going to give it a right good go."
Speaking while fishing with his dad Darren at his local lake in Essex, Barthram said he would be travelling to Cheltenham early next week to find accommodation in the town, ready for the start of pre-season training on Wednesday, July 1.
He was an unused substitute the day Cheltenham thrashed Swindon 5-0 in the FA Cup first round last November.
When asked to describe his game, Barthram said: "I am a solid defender who also likes to attack and create things. I can play on the left as well as the right because I am two footed and I have also played in a holding midfield role.
"Coming through at a club like Tottenham was special and I am looking forward to seeing Jack Munns again. He only lives 10 minutes down the road from me in Essex and I spoke to him before agreeing to join Cheltenham.
"I think having a familiar face in the squad will help us both settle in, but it won't take long for the whole squad to bond."