Ted Cruz, a Republican senator who chairs a subcommittee on aviation and space, said: "I believe it would be prudent for the US likewise to temporarily ground 737 Max aircraft until the FAA confirms the safety of these aircraft and their passengers."
Democratic senators Edward Markey and Richard Blumenthal have written to the FAA - which they referred to as "our aviation safety cop on the beat" - asking that the Boeing 737 Max should be grounded "until the agency can conclusively determine that the aircraft can be operated safely".
Democrat presidential hopeful Elizabeth Warren said that the FAA should follow other nations' lead "immediately" and "get these planes out of the sky".
And Republican senator Mitt Romney said: "Out of an abundance of caution for the flying public, the FAA should ground the 737 Max 8 until we investigate the causes of recent crashes and ensure the plane's airworthiness."
But the FAA said that other civil aviation authorities had not "provided data to us that would warrant action".
Boeing has confirmed that for the past few months it has been developing a "flight control software enhancement" for the aircraft, but says it is confident they are safe to fly.
Airline workers also want the FAA to ground the Boeing 737 Max.
The Association of Flight Attendants-CWA union said it is calling on the FAA "to temporarily ground the 737 MAX fleet in the US out of an abundance of caution".
Its president Sara Nelson, said: "The US has the safest aviation system in the world, but Americans are looking for leadership in this time of uncertainty.
"The FAA must act decisively to restore the public faith in the system.
The Allied Pilots Association told its members: "It is important for you to know that if you feel it is unsafe to work the 737 Max, you will not be forced to fly it."
Southwest Airlines and American Airlines - both major operators of the Boeing 737 Max - are continuing to use the planes.
Southwest Airlines is offering passengers scheduled to fly on one of the Boeing planes the chance to change their bookings.
American Airlines said its "standard policies for changes still apply".
What is a Boeing 737 Max aircraft?
The Boeing 737 Max fleet of aircraft are the latest in the company's successful 737 line. The group includes the Max 7, 8, 9 and 10 models.
By the end of January, Boeing had delivered 350 of the Max 8 model out of 5,011 orders. A small number of Max 9s are also operating.
The Max 7 and 10 models, not yet delivered, are due for roll-out in the next few years.