WASHINGTON—A company owned by Joel Zamel, an Israeli entrepreneur whose work has drawn the scrutiny of special counsel Robert Mueller, formed a strategic partnership with a data firm for President Donald Trump’s campaign in a joint bid to win business from the U.S. government and other clients after the 2016 election, according to people familiar with the matter.

The Psy-Group, one of Mr. Zamel’s firms, signed a memorandum of understanding with Cambridge Analytica LLC, a digital media firm that helped propel Mr. Trump to...