(CNN) Less than a week after President Donald Trump called on Congress to address surprise medical bills, lawmakers in both the House and Senate unveiled bipartisan measures to protect Americans from this costly and frustrating issue.

A bipartisan group of senators, led by Sen. Bill Cassidy, a Louisiana Republican and doctor, rolled out Thursday a bill that would shield people who are seen by out-of-network providers in emergency and certain non-emergency situations. Under the legislation, titled the STOP Surprise Medical Bills Act, consumers would pay the same as they would if they went to an in-network provider.

Hospitals and doctors would automatically receive the difference between patient's payment and the median in-network rate for these services. But insurers and providers could appeal the payment amount to an independent arbitrator.

"Patients should be the reason for the care, not an excuse for the bill," said Cassidy, who has been working on the legislation for nearly a year. "This is a bipartisan solution ensuring patients are protected and don't receive surprise bills that are uncapped by anything but a sense of shame."

Earlier this week, House Energy and Commerce Committee leaders released a draft of The No Surprises Act, which is similar but does not provide for an arbitration process, a method the White House has said it doesn't support.

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