Byram, who can play as a right-back or as a right-winger, joined West Ham from Championship side Leeds United last week.
Snodin, who played for both Leeds and Everton as a midfielder, has revealed what he told Byram before his move to West Ham.
The former Doncaster Rovers midfielder has also stated that he understands why the youngster chose the London outfit over the Toffees.
“I think the opportunity of instant first team football at West Ham was the only thing that stopped Sam Byram from coming to Everton,” Snodin wrote in The Liverpool Echo.
“For me he’s a quality full-back but can also play in several positions. He’d have made a great addition to our squad but when it boils down to it the young boy wants to play football and you can’t blame him.”
“I was confident in my ability but like Sam I’d been doing it a league down in the old Second Division,” added the former Everton man. “Even though I was captain of Leeds and had some good seasons, I was still going into a top team in a higher league.
“When I spoke to Sam I said ‘you thought your decision was hard son, mine was Everton or Liverpool and they were probably one and two in Europe at the time’.”
Byram made his debut for West Ham against Manchester City at the Boleyn Ground in the Premier League on Saturday evening.
The English youngster started the game against City on the substitutes’ bench and came on in the 13th minute when Arsenal-owned right-back Carl Jenkinson suffered an injury.
The 22-year-old played well against Manuel Pellegrini’s side, and had a pass accuracy of 74%, and won one header and one tackle.