The orders would be a bitter defeat for Native American tribes affected by the $3.8 billion Dakota Access line as well as climate activists after months of protests successfully delayed the Dakota project.
Energy Transfer Partners shares rose 2 percent in U.S. trading to $36.45 a share. Oil producers in North Dakota's Bakken region would likely benefit from transporting oil to the U.S. Gulf Coast through the 1,172-mile (1,885 km) pipeline.
The Standing Rock Sioux Tribe, whose reservation is adjacent to the proposed Dakota pipeline, won a key victory when the U.S. Army Corps of Engineers in early December turned down ETP's request for an easement to tunnel under the nearby Missouri River.
Most of the pipeline was complete by the summer of 2016, except for a small section under Lake Oahe, a reservoir that forms part of the river.
The Army later said it would begin an environmental assessment that could delay the project further. It was not clear if Trump's expected order on Tuesday would supercede that move.
Standing Rock officials were not immediately available for comment.
Trump on Monday met with leaders of labor unions, including the Building and Construction trades group and the Laborers International Union of North America, who have been vocal supporters of both pipeline projects.
Both groups had endorsed Trump rival Hillary Clinton in the 2016 U.S. election, but disagreed with her opposition to the Keystone pipeline.
Former President Barack Obama rejected Transcanada Corp's Keystone XL oil pipeline, which would bring Canadian crude from Alberta into the U.S. Gulf, in 2015 after environmentalists campaigned against the project for more than seven years. Transcanada declined to comment.
TransCanada shares rose 1.1 percent to hit their highest since September, but pared gains to last trade up 0.8 percent at C$63.30 on the Toronto Stock Exchange.
Trump's action, which comes in his fourth full day in office, would benefit producers concerned about limited pipeline capacity bringing oil to market from both North Dakota and Canada.
U.S. oil production boomed in recent years and has resulted in fewer foreign imports of crude oil.