Rashford, 18, who is in the 26-man provisional squad for Euro 2016, will play in the warm-up against Australia at the Stadium of Light on Friday.
Barnes - who was 19 when he won the first of 79 England caps in 1983 - also warned of the dangers of putting too much pressure on young players.
"You have to be very careful," he said.
"If young players don't perform we're too quick to write them off."
But the 52-year-old believes Rashford can replicate the form of his breakthrough season at Old Trafford, during which he scored eight goals following his February debut.
"If he plays, he should play with no fear, no pressure," Barnes told BBC Radio 5 live. "He's under no pressure whatsoever. I think he'll do OK because he's handled the Manchester United situation fine."
With fellow striker Daniel Sturridge recovering from a calf injury, Rashford is a likely starter against Australia, in England's second of three warm-up games leading up to Euro 2016 in France.
England manager Roy Hodgson said the youngster would play "a serious part in the game".
He will cut his 26-man squad by three over the weekend, before the 31 May deadline for 23-man squads for Euro 2016 to be submitted.
Manchester United striker Marcus Rashford can "play with no fear" on his England debut, says former winger John Barnes.