Customers with such accounts with taxpayer-backed banking giants Royal Bank of Scotland (RBS) and Lloyds had previously only had limited access to cash machines.
RBS chief executive Ross McEwan said it "wasn't right" that life was being made harder for people who needed help the most.
RBS said that by the end of the year it will re-open access to the Link ATM network for all of its customers. This move will benefit around one million customers with a basic bank account who have been unable to use non-RBS/NatWest cash machines.
This reverses a decision taken in 2011 to limit access to ATMS and is part of RBS's plan to earn back customers' trust.
Mr 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 is also writing to customers on basic accounts to tell them they will get similar freedoms from July. Lloyds declined to say how many of its customers have a basic bank account.
Such accounts are often handed out to people with a poor credit history.