The Republican-controlled House of Representatives has passed a bill to approve the US$8-billion pipeline that would carry bitumen from Alberta to refineries and ports in Texas.
And supporters have said they are confident they'll have the 60 votes needed when the bill goes before the Senate next week.
Brad Wall says there is still a long way to go.
The White House has stopped short of directly threatening to veto the bill.
But it has said the legal right to approve cross-border infrastructure projects belongs to the administration — not to members of Congress.
The latest bid by House Republicans is believed to have the best chance of reaching President Barack Obama's desk.
The pipeline has been stalled by environmental reviews, objections to its route and politics for six years.Suggest a correction