WASHINGTON—The House Intelligence Committee issued seven subpoenas on Wednesday, in a sign that its investigation into alleged Russian meddling in the 2016 election is advancing in scope and intensity, according to people familiar with the matter.

The Republican-led committee issued four subpoenas related to the Russia investigation, targeting President Donald Trump’s former national security adviser Mike Flynn, Mr. Trump’s personal attorney, Michael Cohen, and their businesses. The committee is also investigating possible...