Lloyds says it will also give cash machine freedom to its basic bank account customers.
The accounts provide a stripped-down service, with no cheque book or overdraft.
Adding the whole LINK network will mean customers will have access 67,000 machines which most current account account customers can use.
The U-turn at RBS comes as the new chief executive, Ross McEwan, tries to win back customer confidence.
It restricted access for basic bank accounts in 2011, saying the service was unsustainable because of the cost.
RBS chief executive Ross McEwan said: "We looked at this and decided it just wasn't right.
"You don't make life harder for those who need your help most. We need to rebuild trust with our customers."
Lloyds will start sending out new cards to customers from July, allowing them full cash machine freedom.
RBS will implement the change by the end of the year.