Currently, visitors from 38 mainly stable countries friendly to the United States can make short tourist visits without a visa if they fill in a simple online waiver form before their flight.
But in the wake of last month's attacks in Paris -- carried out by French and Belgian nationals -- both President Barack Obama and his Republican rivals in Congress are looking to crack down.
The administration has already said that henceforth travellers who have visited Syria, Iraq, Somalia and other known militant hotspots would have to declare that fact while registering to visit.
The draft law does not got as far as a measure earlier proposed in the Senate, but it will raise less concern in the US tourism industry and is thus more likely to pass and receive presidential approval.
House speaker Paul Ryan said Representative Candice Miller's bill would come to a vote next week, and Republican majority leader Kevin McCarthy said he fully expected it to become law "as soon as possible."
