WASHINGTON—The Trump administration is moving to erase Obama-era rules on methane emissions from the oil-and-gas business, saying the federal government overstepped its authority when it set limits on what scientists say is a significant contributor to climate change.

The sweeping proposal, formally introduced Wednesday, is the administration’s latest attempt to further boost record crude-oil and natural-gas production by easing regulations. But it comes amid growing concerns over how the industry’s methane emissions are affecting...