Newly disclosed emails shed light on the period leading up to a June 2016 meeting at Trump Tower between a Russian lawyer linked to the Kremlin and top campaign aides to President Donald Trump.

The new information appears to bolster lawyer Natalia Veselnitskaya’s position that she wanted the meeting to argue her case for overturning the Magnitsky Act, a U.S. law targeting Russian human-rights abuses. Ms. Veselnitskaya has previously said the meeting wasn’t organized to provide damaging information about Mr. Trump’s Democratic...